This section introduces multiplicities of zeros for polynomials and goes over the process of accurately graphing a polynomial. Make sure you focus on the local behavior of the function near a zero with multiplicities greater than 2 (discussing the "flattening" as the multiplicity increases). Solving polynomial inequalities using graphs and graphing absolute values of polynomial functions are also required topics. I have included the IVT as an optional topic for those who want to discuss it.
Operational Prerequisites (i.e., "Completing the Homework"): The following prerequisite topics (which have not already been listed as prerequisites in previous sections of this text) are needed to complete the homework.
The following is a list of learning objectives for this section.
Learning Objectives (click to expand)
Recognize characteristics of graphs of polynomial functions.
Use factoring to find zeros of polynomial functions.
Identify zeros and their multiplicities.
Determine end behavior.
Understand the relationship between degree and turning points.
Graph polynomial functions.
Use the Intermediate Value Theorem.
Graph the absolute value of a polynomial function.
Solve a polynomial inequality (using a graph).
The revenue in millions of dollars for a fictional cable company from 2006 through 2013 is shown in Table \(\PageIndex{1}\).
Table \(\PageIndex{1}\)
Year
2006
2007
2008
2009
2010
2011
2012
2013
Revenues
52.4
52.8
51.2
49.5
48.6
48.6
48.7
47.1
The revenue can be modeled by the polynomial function \(R(t)=-0.037 t^4 +1.414 t^3 -19.777 t^2 +118.696t-205.332\), where \(R\) represents the revenue in millions of dollars and \(t\) represents the year, with \(t=6\) corresponding to 2006. Over which intervals is the revenue for the company increasing? Over which intervals is the revenue for the company decreasing?
These questions, along with many others, can be answered by examining the graph of the polynomial function. We have already explored the local behavior of quadratics, a special case of polynomials. In this section we will explore the local behavior of polynomials in general.
Recognizing Characteristics of Graphs of Polynomial Functions
Polynomial functions of degree 2 or more have graphs that do not have sharp corners; recall that these types of graphs are called smooth curves. Polynomial functions also display graphs that have no breaks. Curves with no breaks are called continuous (again, as mentioned in the previous section, this is not technically true; however, we will use this flawed definition of continuity until we reach Calculus). Figure \(\PageIndex{1}\) shows a graph that represents a polynomial function and a graph that represents a function that is not a polynomial.
Figure \(\PageIndex{1}\)
Example \(\PageIndex{1}\): Recognizing Polynomial Functions
Which of the graphs in Figure \(\PageIndex{2}\) represents a polynomial function?
Figure \(\PageIndex{2}\)
Solution
The graphs of \(f\) and \(h\) are graphs of polynomial functions. They are smooth and continuous.
The graphs of \(g\) and \(k\) are graphs of functions that are not polynomials. The graph of function \(g\) has a sharp corner. The graph of function \(k\) is not continuous.
Q&A
Do all polynomial functions have as their domain all real numbers?
Yes. Any real number is a valid input for a polynomial function.
Using Factoring to Find Zeros of Polynomial Functions
Recall that if \(f\) is a polynomial function, the values of \(x\) for which \(f(x)=0\) are called zeros of \(f\). If the equation of the polynomial function can be factored, we can set each factor equal to zero and solve for the zeros.
We can use this method to find \(x\)-intercepts because at the \(x\)-intercepts we find the input values when the output value is zero. For general polynomials, this can be a challenging prospect. While quadratics can be solved using the relatively simple Quadratic Formula, the corresponding formulas for cubic and fourth-degree polynomials are not simple enough to remember, and formulas do not exist for general higher-degree polynomials. Consequently, we will limit ourselves to three cases in this section:
The polynomial can be factored using known methods: greatest common factor and trinomial factoring.
The polynomial is given in factored form.
Technology is used to determine the intercepts.
Example \(\PageIndex{2}\): Finding the \(x\)-Intercepts of a Polynomial Function by Factoring
Find the \(x\)-intercepts of \(f(x)= x^6 -3 x^4 +2 x^2\).
Solution
We can attempt to factor this polynomial to find solutions for \(f(x)=0\).\[\begin{array}{rrclcl} & x^6 -3 x^4 +2 x^2 & = & 0 & & \\[6pt] \implies & x^2(x^4 -3 x^2 +2) & = & 0 & \quad & \left( \text{factoring out the GCF} \right) \\[6pt] \implies & x^2(u^2 -3 u +2) & = & 0 & \quad & \left( \text{substituting }u = x^2\text{ in the trinomial} \right) \\[6pt] \implies & x^2(u - 2)(u - 1) & = & 0 & \quad & \left( \text{factoring the trinomial} \right) \\[6pt] \implies & x^2(x^2 - 2)(x^2 - 1) & = & 0 & \quad & \left( \text{resubstituting }u = x^2 \right) \\[6pt] \implies & x^2(x^2 - 2)(x - 1)(x + 1) & = & 0 & \quad & \left( \text{factoring a Difference of Squares} \right) \\[6pt] \end{array}\nonumber\]Thus,\[\begin{array}{rccccccc} & x^2 = 0 & \text{ or } & x^2 - 2 = 0 & \text{ or } & x - 1 = 0 & \text{ or } & x + 1 = 0 \\[6pt] \implies & x = 0 & & x^2 = 2 & & x = 1 & & x = -1 \\[6pt] \implies & x = 0 & & x = \pm \sqrt{2} & & x = 1 & & x = -1 \\[6pt] \end{array}\nonumber\]This gives us five \(x\)-intercepts: \((0,0)\), \((1,0)\), \((-1,0)\), \((\sqrt{2},0)\), and \((-\sqrt{2},0)\) (see Figure \(\PageIndex{3}\)). We can see that this is an even function.
Figure \(\PageIndex{3}\)
Example \(\PageIndex{3}\): Finding the \(x\)-Intercepts of a Polynomial Function by Factoring
Find the \(x\)-intercepts of \(f(x)= x^3 -5 x^2 -x+5\).
Solution
Find solutions for \(f(x)=0\) by factoring.\[\begin{array}{rrclcl} & x^3 - 5x^2 - x + 5 & = & 0 & & \\[6pt] \implies & x^2(x - 5) - 1 (x - 5) & = & 0 & \quad & \left( \text{factoring by grouping} \right) \\[6pt] \implies & (x - 5)(x^2 - 1) & = & 0 & \quad & \left( \text{factor out the GCF} \right) \\[6pt] \implies & (x - 5)(x - 1)(x + 1) & = & 0 & \quad & \left( \text{factoring the Difference of Squares} \right) \\[6pt] \end{array}\nonumber\]Hence, \(x = 5\), or \(x = 1\), or \(x = -1\). There are three \(x\)-intercepts: \((-1,0)\), \((1,0)\), and \((5,0)\) (see Figure \(\PageIndex{4}\)).
Figure \(\PageIndex{4}\)
Example \(\PageIndex{4}\): Finding the \(y\)- and \(x\)-Intercepts of a Polynomial in Factored Form
Find the \(y\)- and \(x\)-intercepts of \(g(x)= (x-2)^2 (2x+3)\).
Solution
The \(y\)-intercept can be found by evaluating \(g(0)\).\[g(0) = (0-2)^2 (2(0)+3) = 12.\nonumber\]So the \(y\)-intercept is \((0,12)\).
The \(x\)-intercepts can be found by solving \(g(x)=0\).\[(x - 2)^2 (2x + 3) = 0 \implies x = 2 \text{ or } x = -\dfrac{3}{2}.\nonumber\]So the \(x\)-intercepts are \((2,0)\) and \(\left( -\frac{3}{2},0 \right)\).
We can always check that our answers are reasonable by using a graphing calculator to graph the polynomial. For example, Figure \(\PageIndex{5}\) shows that our answers for Example \(\PageIndex{4}\) are correct.
Figure \(\PageIndex{5}\)
Example \(\PageIndex{5}\): Finding the \(x\)-Intercepts of a Polynomial Function Using a Graph
Find the \(x\)-intercepts of \(h(x)= x^3 +4 x^2 +x-6\).
Solution
This polynomial is not in factored form, has no common factors, and does not appear to be factorable using techniques previously discussed. Fortunately, we can use technology to find the intercepts. Keep in mind that some values make graphing difficult by hand. In these cases, we can take advantage of graphing utilities.
Looking at the graph of this function, as shown in Figure \(\PageIndex{6}\), it appears that there are \(x\)-intercepts at \(x=-3\), \(-2\), and \(1\).
Figure \(\PageIndex{6}\)
We can check whether these are correct by substituting these values for \(x\) and verifying that\[h(-3)=h(-2)=h(1)=0.\nonumber\]Since \(h(x)= x^3 +4 x^2 +x-6\), we have:\[\begin{array}{rcl} h(-3) & = & (-3)^3 + 4(-3)^2 + (-3) - 6 = -27 + 36 - 3 - 6 = 0 \\[6pt] h(-2) & = & (-2)^3 + 4(-2)^2 + (-2) - 6 = -8 + 16 - 2 - 6 = 0 \\[6pt] h(1) & = & (1)^3 + 4(1)^2 + (1) - 6 = 1 + 4 + 1 - 6 = 0 \\[6pt] \end{array}\nonumber\]Each \(x\)-intercept corresponds to a zero of the polynomial function and each zero yields a factor, so we can now write the polynomial in factored form.\[h(x)= x^3 +4 x^2 +x-6 =(x+3)(x+2)(x-1).\nonumber\]
Checkpoint \(\PageIndex{5}\)
Find the \(y\)- and \(x\)-intercepts of the function \(f(x)= x^4 -19 x^2 +30x\).
Identifying Zeros and Their Multiplicities
Graphs behave differently at various \(x\)-intercepts. Sometimes, the graph will cross over the horizontal axis at an intercept. Other times, the graph will touch the horizontal axis and bounce off.
Suppose, for example, we graph the function \(f(x)=(x+3) (x-2)^2 (x+1)^3\). Notice in Figure \(\PageIndex{7}\) that the behavior of the function at each of the \(x\)-intercepts is different.
Figure \(\PageIndex{7}\):Identifying the behavior of the graph at an \(x\)-intercept by examining the multiplicity of the zero.
The \(x\)-intercept at \(x = -3\) is the solution of equation \((x+3)=0\). The graph passes directly through the \(x\)-intercept at \(x=-3\). We say the factor is linear (has a degree of 1), so the behavior near the intercept is like that of a line—it passes directly through the intercept. We call this a single zero because the zero corresponds to a single factor of the function.
The \(x\)-intercept at \(x = 2\) is the repeated solution of equation \((x-2)^2 = 0\). The graph touches the axis at the intercept and changes direction. We say the factor is quadratic (degree 2), so the behavior near the intercept is like that of a quadratic—it bounces off of the horizontal axis at the intercept. Since \((x-2)^2 =(x-2) (x-2)\), the factor is repeated. This leads us to an important definition that will be useful when it comes to polynomial functions.
Definition: Multiplicity of a Zero
Let \(h\) be a zero of a nonzero polynomial function \(f\). The multiplicity of \(h\) is the largest positive integer \(p\) for which \((x-h)^p\) is a factor of \(f(x)\). We say that \(x=h\) is a zero of multiplicity \(p\). A zero of multiplicity \(1\) is called a simple zero; a zero of multiplicity greater than \(1\) is called a repeated zero (or multiple zero).
Hence, the zero associated with \(x=2\) has multiplicity 2 because the factor \((x-2)\) occurs twice.
The \(x\)-intercept at \(x = -1\) is the repeated solution of factor \((x+1)^3 = 0\). The graph passes through the axis at the intercept, but flattens out a bit first. We say this factor is cubic (degree 3), so the behavior near the intercept is like that of a cubic—with the same "S"-shape near the intercept as the toolkit function \(f(x)= x^3\). By definition, this is a zero of multiplicity 3.
Now that we know what it means for a zero to have multiplicity \(p\), it is natural to ask if there is any ramification of the multiplicity of a zero.
Theorem: Local Behavior Near a Zero
If \( x = h \) is a zero of multiplicity \( p \) for a given polynomial, then the local behavior of the graph of the polynomial near \( x = h \) matches the behavior of \( y = \pm (x - h)^p \), where the sign of the coefficient is determined by other characteristics of the given polynomial.
In truth, most people (myself included) would say that the behavior of the polynomial near to a zero of multiplicity \(p\) matches the graph of \(y=x^p\), but possibly reflected about the \(x\)-axis. What we mean is that the behavior of the graph near to the zero matches the look of \(y = x^p\) near the origin.
For zeros with even multiplicities, the graphs touch or are tangent to the \(x\)-axis. For zeros with odd multiplicities, the graphs cross or intersect the \(x\)-axis. See Figure \(\PageIndex{8}\) for examples of graphs of polynomial functions with multiplicity \(1\), \(2\), and \(3\).
Figure \(\PageIndex{8}\)
For higher even powers, such as \(4\), \(6\), and \(8\), the graph will still touch and bounce off of the horizontal axis but, for each increasing even power, the graph will appear flatter as it approaches and leaves the \(x\)-axis. For higher odd powers, such as \(5\), \(7\), and \(9\), the graph will still cross through the horizontal axis, but for each increasing odd power, the graph will appear flatter as it approaches and leaves the \(x\)-axis.
Interactive Element: Multiplicity and the Shape of a Graph Near a Zero
The following Interactive Element might help you visualize how the multiplicity of a zero controls whether the graph crosses or bounces, and how it flattens near the zero.
Interact: Drag the red handle left and right to change the multiplicity \(n\) of the zero at the origin.
Observation: Odd \(n\) makes the graph cross the \(x\)-axis, while even \(n\) makes it touch and bounce; and as \(n\) grows, the graph flattens more near the zero — just like \(y=x^n\).
Caution: Graph Accurately Near Zeros!
You will eventually be asked to make accurate sketches of polynomial functions. Part of this skill is to show that you know the impact of high multiplicity on a zero. The higher the multiplicity, the flatter the polynomial graph is near to the zero—be prepared to show this.
Finally, you will inevitably notice the following fact.
Theorem: Sum of Multiplicities and the Degree of a Polynomial
The degree of a polynomial factored into the product of linear and irreducible quadratic factors is sum of the multiplicities of the linear factors and the twice the sum of the multiplicities of the irreducible quadratic factors. That is, if\[ P(x) = (a_1 x + b_1)^{m_1} (a_2 x + b_2)^{m_2} \cdots (a_k x + b_k)^{m_k} (c_1 x^2 + d_1)^{p_1} (c_2 x^2 + d_2)^{p_2} \cdots (c_n x^2 + d_n)^{p_n}, \nonumber \]then the degree of \( P(x) \) is\[ m_1 + m_2 + \cdots + m_k + 2(p_1 + p_2 + \cdots p_n). \nonumber \]
Example \(\PageIndex{6}\): Identifying Zeros and Their Multiplicities
Use the graph of the function of degree 6 in Figure \(\PageIndex{9}\) to identify the zeros of the function and their possible multiplicities.
Figure \(\PageIndex{9}\)
Solution
The polynomial function is of degree \(n\). The sum of the multiplicities must be \(n\).
Starting from the left, the first zero occurs at \(x=-3\). The graph touches the \(x\)-axis, so the multiplicity of the zero must be even. The zero of \(-3\) could have multiplicity \(2\), \(4\), or \(6\) (remember, we were told this polynomial has degree 6, so the sum of the multiplicities of the zeros must be 6).
The next zero occurs at \(x=-1\). The graph looks almost linear at this point, so this is likely a zero of multiplicity 1.
The last zero occurs at \(x=4\). The graph crosses the \(x\)-axis, so the multiplicity of the zero must be odd. From the shape of the curve near to \(x = 4\), we know the multiplicity cannot be \(1\) (the graph is not locally linear near \(x = 4\)). Hence, \(x = 4\) must be a zero of multiplicity either \(3\) or \(5\).
Now that we have an idea of the possible multiplicities, let's build a table and use a little deductive logic to identify the exact multiplicities.\[\begin{array}{cc} \textbf{Zero} & \textbf{Possible Multiplicities} \\[6pt] \hline -3 & 2,4,6 \\[6pt] -1 & 1 \\[6pt] 4 & 3,5 \\[6pt] \end{array}\nonumber\]Since the sum of the multiplicities must be 6, and we have three zeros, the maximum multiplicity for any given zero is 3. This cleans up our table quite a bit:\[\begin{array}{cc} \textbf{Zero} & \textbf{Possible Multiplicities} \\[6pt] \hline -3 & 2 \\[6pt] -1 & 1 \\[6pt] 4 & 3 \\[6pt] \end{array}\nonumber\]Thus, we have identified the zeros and their exact multiplicities.
Checkpoint \(\PageIndex{6}\)
Use the graph of the function of degree 7 in Figure \(\PageIndex{10}\) to identify the zeros of the function and their multiplicities.
Figure \(\PageIndex{10}\)
Determining End Behavior
As we have already learned, the behavior of a graph of a polynomial function of the form\[f(x)= a_n x^n + a_{n-1} x^{n-1} + \ldots + a_1 x + a_0\nonumber\]will either ultimately rise or fall as \(x\) increases or decreases without bound. This is because, for very large inputs, say 100 or 1,000, the leading term dominates the size of the output. The same is true for very negative inputs, say –100 or -1,000.
Recall that we call this behavior the end behavior of a function. As we pointed out when discussing power functions, when the leading term of a polynomial function, \(a_n x^n\), is an even power function, as \(x\) increases or decreases without bound, \(f(x)\) increases without bound. When the leading term is an odd power function, as \(x\) decreases without bound, \(f(x)\) also decreases without bound; as \(x\) increases without bound, \(f(x)\) also increases without bound. If the leading term is negative, it will change the direction of the end behavior. Figure \(\PageIndex{11}\) summarizes all four cases.
Figure \(\PageIndex{11}\)
Understanding the Relationship Between Degree and Turning Points
In addition to the end behavior, recall that we can analyze a polynomial function's local behavior. It may have a turning point where the graph changes from increasing to decreasing (rising to falling) or decreasing to increasing (falling to rising). Look at the graph of the polynomial function \(f(x)= x^4 - x^3 -4 x^2 +4x\) in Figure \(\PageIndex{12}\). The graph has three turning points.
Figure \(\PageIndex{12}\)
This function is a \(4^{\text{th}}\) degree polynomial function and has 3 turning points. This should be familiar because, as mentioned in the previous section, the maximum number of turning points of a polynomial function is always one less than the degree of the function.
Example \(\PageIndex{7}\): Finding the Maximum Number of Turning Points Using the Degree of a Polynomial Function
Find the maximum number of turning points of each polynomial function.
\(f(x) = -x^3 + 4 x^5 - 3 x^2 +1\)
\(f(x) = -(x-1)^2 (1 + 2x^2)\)
Solutions
First, rewrite the polynomial function in descending order: \(f(x) = 4x^5 - x^3 - 3x^2 + 1\). Identify the degree of the polynomial function. This polynomial function is of degree 5. The maximum number of turning points is \(5-1=4\).
First, identify the leading term of the polynomial function if the function were expanded. Looking at the given function, \(f(x) = -(x-1)^2 (1 + 2x^2)\), we can see that the factor \((x - 1)^2\) will contribute \(x^2\) (if we multiplied it out) and the factor \(1 + 2x^2\) contributes \(2x^2\). Therefore, the leading term is\[a_n x^n = - (x^2) (2x^2) = -2 x^4.\nonumber\]Therefore, this polynomial function is of degree 4. As such, the maximum number of turning points is \(4-1=3\).
Graphing Polynomial Functions
We can use what we have learned about multiplicities, end behavior, and turning points to sketch graphs of polynomial functions. Let us put this all together and look at the steps required to graph polynomial functions.
How To: Graph a polynomial function
Find the intercepts.
Check for symmetry. If the function is an even function, its graph is symmetric about the \(y\)-axis. That is, \(f(-x)=f(x)\). If a function is an odd function, its graph is symmetric about the origin. That is, \(f(-x)=-f(x)\).
Use the multiplicities of the zeros to determine the behavior of the polynomial at the \(x\)-intercepts.
Determine the end behavior by examining the leading term.
Use the end behavior and the behavior at the intercepts to sketch a graph.
Ensure that the number of turning points does not exceed one less than the degree of the polynomial.
Optionally, use technology to check the graph.
Example \(\PageIndex{8}\): Sketching the Graph of a Polynomial Function
Sketch a graph of \(f(x) = -2(x+3)^2 (x-5)\).
Solution
This graph has two \(x\)-intercepts. At \(x=-3\), the factor is squared, indicating a multiplicity of 2. The graph will bounce at this \(x\)-intercept. At \(x=5\), the function has a multiplicity of one, indicating the graph will cross through the axis like a line at this intercept.
The \(y\)-intercept is found by evaluating \(f(0)\).\[f(0) = -2(0+3)^2(0-5) = -2 \cdot 9 \cdot (-5) = 90.\nonumber\]The \(y\)- intercept is \((0,90)\).
Additionally, we can see the leading term, if this polynomial were multiplied out, would be \(-2 x^3\), so the end behavior is that of a vertically reflected cubic, with the outputs decreasing as the inputs approach infinity, and the outputs increasing as the inputs approach negative infinity. See Figure \(\PageIndex{13}\).
Figure \(\PageIndex{13}\)
To sketch this, we consider that:
As \(x \to - \infty\), \(f(x) \to \infty\). So we know the graph starts in the second quadrant and is decreasing toward the \(x\)-axis.
Since \(f(-x) = -2 (-x+3)^2 (-x-5)\) is not equal to \(\pm f(x)\), the graph does not display symmetry.
At \((-3,0)\), the graph bounces off of the \(x\)-axis, so the function must start increasing.
At \((0,90)\), the graph crosses the \(y\)-axis at the \(y\)-intercept. See Figure \(\PageIndex{14}\).
Figure \(\PageIndex{14}\)
Somewhere after this point, the graph must turn back down or start decreasing toward the horizontal axis because the graph passes through the next intercept at \((5,0)\). See Figure \(\PageIndex{15}\).
Figure \(\PageIndex{15}\)
As \(x \to \infty\), the function \(f(x) \to - \infty\), so we know the graph continues to decrease, and we can stop drawing the graph in the fourth quadrant.
Using technology, we can create the graph for the polynomial function, shown in Figure \(\PageIndex{16}\), and verify that the resulting graph looks like our sketch in Figure \(\PageIndex{15}\).
Figure \(\PageIndex{16}\):The complete graph of the polynomial function \(f(x) = -2(x+3)^2 (x-5)\)
Checkpoint \(\PageIndex{8}\)
Sketch a graph of \(f(x)= \frac{1}{4} x (x-1)^4 (x+3)^3\).
While you will not be graphing the absolute value of polynomial functions in Calculus, you will be graphing the absolute value of other functions. Therefore, it's a good idea to keep that skill fresh (it's a simple skill once you know how to graph the non-absolute value function).
Example \(\PageIndex{9}\): Revisiting the Graphs of Absolute Value Functions
As we have done in previous sections, we initially concern ourselves with the non-absolute value function\[f(x) = -0.9(x + 2)^3 (x - 1)^6(x - 5)^5.\nonumber\]We list the roots and their multiplicities, as well as the behavior of the graph near that root:\[\begin{array}{c|c|c}
\textbf{Root} & \textbf{Multiplicity}& \text{Behavior}\\[6pt]
\hline-2 & 3 & \text{like }x^3 \\[6pt]
1 & 6 & \text{like }x^6 \text{ (a very flat parabola)} \\[6pt]
5 & 5 & \text{like }x^5 \text{ (a flat cubic)} \\[6pt]
\end{array}\nonumber\]Next, we note that the degree of the polynomial is \(3 + 6 + 5 = 14\), so the end behavior is like that of \(x^2\) except the leading coefficient is negative. Therefore, the end behavior is more like \(-x^2\). That is, as \(x \to -\infty\), \(f(x) \to -\infty\), and as \(x \to \infty\), \(f(x) \to -\infty\). Reading from left to right, the graph of \(f\) should
start low
increase until it crosses the \(x\)-axis at \(x = -2\) like a cubic
climb for a bit, but then turn to fall back to the \(x\)-axis to "bounce" like a very flat parabola at \(x = 1\)
climb for a bit, but then turn to fall back to the \(x\)-axis to cross at \(x = 5\) like a flat cubic
it then continues to fall forever.
Figure \(\PageIndex{17}\) shows the graph of the non-absolute value function.
Figure \(\PageIndex{17}\)
To graph the absolute value of this function, we take all the function outputs (\(y\)-values) that are negative and we force them to be positive. In effect, this keeps the part of the graph from Figure \(\PageIndex{17}\) that is already above the \(x\)-axis in place; however, all the parts that are below the \(x\)-axis get reflected about the \(x\)-axis to become positive (above the \(x\)-axis). See Figure \(\PageIndex{18}\).
Figure \(\PageIndex{18}\)
Solving Polynomial Inequalities
We finish this section with a topic that might not seem related to graphing polynomial functions—solving polynomial inequalities.
There are those who insist on students solving inequalities using sign charts, and then there are those who want to reinforce a student's graphing skills. If you cannot tell, I am in the latter camp. To solve any inequality, you could move all terms to one side of the inequality, graph the resulting expression on the non-zero side, and use that graph to solve the inequality. The key to being able to do this is know how to graph that non-zero expression.
Let's see this in an example.
Example \(\PageIndex{10}\)
Solve\[3x^3 -4x \geq6x^2 - 8.\nonumber\]
Solution
Let's start by getting all terms to one side.\[\begin{array}{rrcl}
& 3x^3 -4x & \geq & 6x^2 - 8 \\[6pt]
\implies & 3x^3 - 6x^2 - 4x + 8 & \geq & 0\\[6pt]
\end{array}\nonumber\]Now we want to graph the expression on the left side. That is, let's graph\[g(x) =3x^3 - 6x^2 - 4x + 8.\nonumber\]Since this is a polynomial, we would like to find the roots and their multiplicities. Let's try factoring by grouping:\[\begin{array}{rcl}
g(x) & = & 3x^3 - 6x^2 - 4x + 8 \\[6pt]
& = & 3x^2(x- 2) - 4(x -2)\\[6pt]
& = & (x - 2)(3x^2- 4)\\[6pt]
\end{array}\nonumber\]Hence, this polynomial has the following roots (and multiplicities):\[\begin{array}{c|c}
\textbf{Root} & \textbf{Multiplicity} \\[6pt]
\hline2 & 1 \\[6pt]
\dfrac{2}{\sqrt{3}} & 1 \\[6pt]
-\dfrac{2}{\sqrt{3}} & 1 \\[6pt]
\end{array}\nonumber\]Since \(g(x)\) is a cubic, we know it "starts low and ends high." A quick sketch yields the graph in Figure \(\PageIndex{19}\).
Figure \(\PageIndex{19}\)
Remember, we are being asked to solve \(3x^3 - 6x^2 - 4x + 8 \geq 0\). This is equivalent to asking, "When is the graph of \(g(x) =3x^3 - 6x^2 - 4x + 8\) at or above the \(x\)-axis?" The answer to this question is simple once you have the graph. We can see from Figure \(\PageIndex{19}\), that \(g(x) \geq 0\) when\[x \in \left[ -\dfrac{2}{\sqrt{3}},\dfrac{2}{\sqrt{3}} \right] \cup \left[ 2,\infty \right).\nonumber\]
Using the Intermediate Value Theorem (optional topic)
In some situations, we may know two points on a graph but not the zeros. If those two points are on opposite sides of the \(x\)-axis, we can confirm that there is a zero between them. Consider a polynomial function \(f\) whose graph is smooth and continuous. The Intermediate Value Theorem states that for two numbers \(a\) and \(b\) in the domain of \(f\), if \(a < b\) and \(f(a) \neq f(b)\), then the function \(f\) takes on every value between \(f(a)\) and \(f(b)\). We can apply this theorem to a special case that is useful in graphing polynomial functions. If a point on the graph of a continuous function \(f\) at \(x=a\) lies above the \(x\)-axis and another point at \(x=b\) lies below the \(x\)-axis, there must exist a third point between \(x=a\) and \(x=b\) where the graph crosses the \(x\)-axis. Call this point \((c,f(c))\). This means that we are assured there is a solution \(c\) where \(f(c)=0\).
In other words, the Intermediate Value Theorem tells us that when a polynomial function changes from a negative value to a positive value, the function must cross the \(x\)-axis. Figure \(\PageIndex{20}\) shows that there is a zero between \(a\) and \(b\).
Figure \(\PageIndex{20}\):Using the Intermediate Value Theorem to show there exists a zero.
Theorem: Intermediate Value Theorem (for Polynomials)
Let \(f\) be a polynomial function. If \(f( a )\) and \(f( b )\) have opposite signs, then there exists at least one value \(c\) between \(a\) and \(b\) for which \(f( c )=0\).
Proof
This proof must wait until Calculus.
To be honest, the Intermediate Value Theorem applies to any continuous function (not just polynomials); however, that extra bit of power to the theorem is reserved for Calculus (where the Intermediate Value Theorem becomes incredibly useful).
Example \(\PageIndex{11}\): Using the Intermediate Value Theorem
Show that the function \(f(x)= x^3 -5 x^2 + 3x +6\) has at least two real zeros between \(x=1\) and \(x=4\).
Solution
As a start, evaluate \(f(x)\) at the integer values \(x=1\), \(2\), \(3\), and \(4\). See Table \(\PageIndex{2}\).
Table \(\PageIndex{2}\)
\(x\)
1
2
3
4
\(f(x)\)
5
0
–3
2
We see that one zero occurs at \(x=2\). Also, since \(f(3)\) is negative and \(f(4)\) is positive, by the Intermediate Value Theorem, there must be at least one real zero between 3 and 4. Thus, we have shown that there are at least two real zeros between \(x=1\) and \(x=4\).
We can verify this result by looking at the graph of the function in Figure \(\PageIndex{21}\).
Figure \(\PageIndex{21}\)
Checkpoint \(\PageIndex{11}\)
Show that the function \(f(x)=7 x^5 -9 x^4 - x^2\) has at least one real zero between \(x=1\) and \(x=2\).
Using Local and Global Extrema
With quadratics, we were able to algebraically find the maximum or minimum value of the function by finding the vertex. For general polynomials, finding these turning points is not possible without more advanced techniques from Calculus. Even then, finding where extrema occur can still be algebraically challenging. For now, we will estimate the locations of turning points using technology to generate a graph.
Each turning point represents a local minimum or maximum. We restate the definition from earlier in this textbook below for completeness.
Definition: Local Extrema
A function \( f \) has a local maximum at \( x= b \) in an open interval \( ( a , c ) \) if \( f ( b ) \geq f(x) \) for all \(x \neq b\) in the interval. Likewise, \( f \) has a local minimum at \( x = b \) in \( ( a , c ) \) if \( f ( b ) \leq f(x) \) for all \( x \neq b \) in the interval.
If \( f \) has more than one local maximum on a given interval, we call the collection of these maximums the local maxima. Likewise, if \( f \) has more than one local minimum on a given interval, we call the collection of these minimums the local minima.
Together, local maxima and minima are called local extrema (or local extreme values).
Some texts use the word "relative" instead of "local." We will use the latter throughout this text.
Sometimes, a turning point is the highest or lowest point on the entire graph. In these cases, we say that the turning point is a global maximum or a global minimum. Again, we restate the definition from earlier in this text below for completeness.
Definition: Absolute Extrema
A function \( f \) has an absolute maximum at \( x = b \) if \( f ( b ) \geq f ( x ) \) for all \( x \) in the domain of \( f \). Likewise, a function \( f \) has an absolute minimum at \( x = b \) if \( f ( b ) \leq f ( x ) \) for all \( x \) in the domain of \( f \).
If \( f \) has more than one local absolute maximum, we call the collection of these maximums the absolute maxima. Likewise, if \( f \) has more than one absolute minimum, we call the collection of these minimums the absolute minima.
Together, absolute maxima and minima are called absolute extrema.
Some texts use the word "global" instead of "absolute." We will use the latter throughout this text.
We can see the difference between local and global extrema in Figure \(\PageIndex{22}\).
Figure \(\PageIndex{22}\)
Q&A
Do all polynomial functions have a global minimum or maximum?
No. Only polynomial functions of even degree have a global minimum or maximum. For example, \(f(x)=x\) has neither a global maximum nor a global minimum.
Example \(\PageIndex{12}\): Using Local Extrema to Solve Applications
An open-top box is to be constructed by cutting out squares from each corner of a 14 cm by 20 cm sheet of plastic then folding up the sides. Find the size of squares that should be cut out to maximize the volume enclosed by the box.
Solution
We will start this problem by drawing a picture like that in Figure \(\PageIndex{23}\), labeling the width of the cut-out squares with a variable, \(w\).
Figure \(\PageIndex{23}\)
Notice that after a square is cut out from each end, it leaves a \((14 - 2w)\) cm by \((20 - 2w)\) cm rectangle for the base of the box, and the box will be \(w\) cm tall. This gives the volume\[V(w) = (20-2w)(14-2w)w = 280w - 68w^2 + 4w^3.\nonumber\]Notice, since the factors are \(w\), \(20-2w\), and \(14-2w\), the three zeros are \(0\), \(10\), and \(7\), respectively. Because a height of 0 cm is not reasonable, we consider only the zeros 10 and 7. The shortest side is 14 and we are cutting off two squares, so values \(w\) may take on are greater than zero or less than 7. This means we will restrict the domain of this function to \(0<w<7\). Using technology to sketch the graph of \(V(w)\) on this reasonable domain, we get a graph like that in Figure \(\PageIndex{24}\). We can use this graph to estimate the maximum value for the volume, restricted to values for \(w\) that are reasonable for this problem—values from 0 to 7.
Figure \(\PageIndex{24}\)
From this graph, we turn our focus to only the portion on the reasonable domain, \([0,7]\). We can estimate the maximum value to be around 340 cubic cm, which occurs when the squares are about 2.75 cm on each side. To improve this estimate, we could use advanced features of our technology, if available, or simply change our window to zoom in on our graph to produce Figure \(\PageIndex{25}\).
Figure \(\PageIndex{25}\)
From this zoomed-in view, we can refine our estimate for the maximum volume to about 339 cubic cm, when the squares measure approximately 2.7 cm on each side.
Checkpoint \(\PageIndex{12}\)
Use technology to find the maximum and minimum values on the interval \([-1,4]\) of the function \(f(x) = -0.2(x-2)^3 (x+1)^2 (x-4)\).
Success in College: Identifying Your Learning Style(s)
Visual/Verbal Learners
These people learn best when information is presented visually and in a written form. In a classroom setting, they prefer professors who use visual aids (i.e., whiteboard or PowerPoint presentation) to list the essential points of a lecture to provide them with an outline to follow. They benefit from information obtained from textbooks and class notes. These learners like to study by themselves in quiet environments. They visualize information in their "minds' eyes" to remember something. The online environment is especially appropriate for visual/verbal learners because most of the information for a course is presented in written form.
Visual/Nonverbal Learners
These people learn best when information is presented visually and in a picture or design format. In a classroom setting, they benefit from professors who supplement their lectures with materials such as film, video, maps, and diagrams. They relate well to information obtained from the images and charts in textbooks. They tend to prefer to work alone in quiet environments. They visualize an image of something in their mind when trying to remember it. These learners may also be artistic and enjoy visual art and design. The online environment is well suited for this learner because graphical representations of information can help them remember concepts and ideas. Graphical information can be presented using charts, tables, graphs, and images.
Auditory/Verbal Learners
These people learn best when information is presented aurally. They benefit from listening to lectures and participating in group discussions in a classroom setting. They also benefit from obtaining information from audio recordings. When trying to remember something, they often repeat it aloud and mentally "hear" how the professor explained the information to them. They learn best when interacting with others in a listening/speaking activity. Online learning environments can complement these learners' styles. Although most information is presented visually (either written or graphically), group participation and collaborative activities are accomplished well online. In addition, streaming audio and computer conferencing can be incorporated into an online course to best meet the learning style of these students.
Tactile/Kinesthetic Learners
These people learn best when doing a physical "hands-on" activity. In the classroom, they prefer to learn new materials in a lab setting where they can touch and manipulate materials. They learn best in physically active learning situations. They benefit from professors who use in-class demonstrations, hands-on learning experiences, and fieldwork outside the classroom. Online environments can provide learning opportunities for tactile/kinesthetic learners. Simulations with 3-Dimensional graphics can replicate physical demonstrations. Lab sessions can be conducted at predetermined locations or homes and then discussed online. Also, outside fieldwork can be incorporated into the coursework, with ample online discussion preceding and following the experience. Finally, the online environment is well suited for the presentation and discussion of either group or individual projects and activities.
What was your most extreme score, and what was it about?
Of all the results, which surprised you the most?
Success in Mathematics: Math Anxiety (Part 1)
The following is an excerpt of an article written at MathAcademy.com.
A famous stage actress was once asked if she had ever suffered from stage fright and, if so, how she had gotten over it. She laughed at the interviewer's naive assumption that she must not feel that kind of anxiety since she was an accomplished actress. She assured him that she had always had stage fright and had never gotten over it. Instead, despite it, she had learned to walk on stage and perform.
Like stage fright, math anxiety can be a disabling condition, causing humiliation, resentment, and even panic.
Even though we all suffer from math anxiety to some degree – just as anyone feels at least a little nervous when speaking to an audience – for some of us, it is a severe problem. This burden interferes with our lives, preventing us from achieving our goals. The first step, without which no further progress is possible, is to recognize that math anxiety is an emotional response. (In fact, severe math anxiety is a learned emotional response.) As with any strong emotional reaction, there are constructive and unconstructive ways to manage math anxiety. Unconstructive (and damaging) ways include rationalization, suppression, and denial.
By "rationalization," we mean finding reasons why it is okay and perhaps even inevitable – and therefore justified – for you to have this reaction. The myths of "men are better at math than women" or "math is for logical people and not creative people" are examples of rationalizations. While they may make you feel better (or at least less bad) about having math anxiety, they will do nothing to lessen it or to help you get it under control. Therefore, rationalization is unconstructive.
"Suppression" means being aware of the anxiety – but trying very hard not to feel it. I have found that students very commonly attempt this, and some severe self-criticism accompanies it. Students feel that they shouldn't feel this anxiety, that it's a weakness they should overcome by brute force if necessary. When this effort doesn't succeed (as invariably it doesn't), the self-criticism becomes ever harsher, leading to a deep sense of frustration and often a severe loss of self-esteem – mainly if the stakes for a student are high, as when their career or personal goals are riding on a successful outcome in a math class, or when parental disapproval is a factor. Consequently, suppression of math anxiety is not only unconstructive but can be damaging.
Finally, there is denial. People using this approach probably aren't likely to see this essay, much less read it, for they carefully construct their lives to avoid all mathematics as much as possible. They choose college majors and careers that don't require math and let the bank or their spouse balance the checkbook. This approach has the advantage that feelings of frustration and anxiety about math are mostly avoided. However, their lives are drastically constrained, for, in our society, fewer than 25% of all careers are, so to speak, "math-free." Thus, their choices of personal and professional goals are severely limited. (Incidentally, most math-free jobs are low-status and low-pay.)
People in denial about mathematics miss out on something else, too, for the student of mathematics learns to see aspects of the structure and beauty of our world that can be seen in no other way and to which the "innumerate" necessarily remain forever blind. It would be a lot like never hearing music or never seeing colors. (I understand that some people have these disabilities but didn't choose to have them.)
Okay, so what is the constructive way to manage math anxiety? I call it "taking possession." It involves identifying the sources of math anxiety in one's life, accepting those feelings without self-criticism, and then learning strategies for disarming math anxiety's influence on one's future study of mathematics.
Begin by understanding that your feelings of math anxiety are not uncommon and do not indicate that there is anything wrong with you or inferior about your ability to learn math. For some, this can be hard to accept, but it is worth trying to accept – since, after all, it happens to be true. This can be made more accessible by exploring your own "math history." Think back across your career as a math student, and identify those experiences which have contributed most to your feelings of frustration about math. For some, this will be a memory of a humiliating experience in school, such as being made to stand at the blackboard and embarrassed in front of one's peers. For others, it may involve interaction with a parent. Whatever the principle episodes are, recall them as vividly as you can. Then, write them down. This is important. After you have written the episode on a sheet(s) of paper, write down your reaction to the episode, both at the time and how it makes you feel to recall it now.
After completing this exercise, take a fresh sheet of paper and try to sum up in a few words what your feelings about math are at this point in your life, together with the reason or reasons you wish to succeed at math. This, too, is important. Not until after we lay out for ourselves consciously and deliberately what our feelings and desires are towards mathematics will it become possible to take possession of our feelings of math anxiety and become free to implement strategies for coping with those feelings.
At this point, sharing your memories, feelings, and goals with others can be enormously helpful. In a math class I teach for art majors, I hand out a questionnaire early in the semester asking students to do precisely what is described above. After spending about twenty minutes writing down their recollections and goals, I lead them in a classroom discussion on math anxiety. This process of dialogue and sharing – though it may seem just a bit on the goopy side – invariably brings out of each student their barriers to math, often helping them become completely conscious of these barriers for the first time. Just as important, it helps all my students understand that the negative experiences they have had, and their reactions to them, are shared one way or another by almost everyone else in the room.
If you do not have the opportunity to engage in a group discussion in a classroom setting, find friends or relatives you trust to respect your feelings and induce them to talk about their own math anxiety experiences and listen to yours.
Here are some questions about this topic that may help you succeed in this class (and possibly many other classes as well):
Are there counselors at your school that are available to help students who are struggling with math anxiety? If so, where are they located, and how to make an appointment to see one?
Estimate the percent of students in your class you think are negatively affected by math anxiety.
If you are unsure about the answers to either of these questions, feel free to ask your instructor - that's what they are there for!
