1.4: First steps in graph sketching

Learning Objectives

1. Identify even and odd functions based on their graph.
2. Determine algebraically whether a function is even or odd.
3. Sketch the graph of a simple polynomial of the form \(y=a x^{n}+b x^{m}\).
4. Sketch a rational function such as \(y=A x^{n} /\left(b+x^{m}\right)\).

Even and Odd Power Functions

So far, we have considered power functions \(y=x^{n}\) with \(x>0\). But in mathematical generality, there is no reason to restrict the independent variable \(x\) to positive values. Thus we expand the discussion to consider all real values of \(x\). We examine now some symmetry properties that arise.

Adjust the slider to see how the even and odd power functions behave as their power increases.

Figure 1.4: Graphs of power functions. (a) A few even power functions: \(y=x^{2}\), \(y=x^{4}\) and \(y=x^{6}\). (b) Some odd power functions: \(y=x, y=x^{3}\) and \(y=x^{5}\). Note the symmetry properties.

In Figure \(1.4\) (a) we see that power functions with an even power, such as \(y=x^{2}, y=x^{4}\), and \(y=x^{6}\), are symmetric about the \(y\)-axis. In Figure 1.4(b) we notice that power functions with an odd power, such as \(y=x, y=x^{3}\) and \(y=x^{5}\) are symmetric when rotated \(180^{\circ}\) about the origin. We adopt the term even function and odd function to describe such symmetry properties. More formally,

\[\begin{aligned} f(-x)=f(x) & \Rightarrow \quad \mathrm{f} \text { is an even function, } \\ f(-x)=-f(x) & \Rightarrow \quad \mathrm{f} \text { is an odd function } \end{aligned} \nonumber \]

Many functions are not symmetric at all, and are neither even nor odd. See Appendix \(\mathrm{C}\) for further details.

Example 1.6

Show that the function \(y=g(x)=x^{2}-3 x^{4}\) is an even function

Solution

For \(g\) to be an even function, it should satisfy \(g(-x)=g(x)\). Let us calculate \(g(-x)\) and see if this requirement holds. We find that

\[g(-x)=(-x)^{2}-3(-x)^{4}=x^{2}-3 x^{4}=g(x) . \nonumber \]

Here we have used the fact that \((-x)^{n}=(-1)^{n} x^{n}\), and that when \(n\) is even, \((-1)^{n}=1\).

All power functions are continuous and unbounded: for \(x \rightarrow \infty\) both even and odd power functions satisfy \(y=x^{n} \rightarrow \infty\). For \(x \rightarrow-\infty\), odd power functions tend to \(-\infty\). Odd power functions are one-to-one: that is, each value of \(y\) is obtained from a unique value of \(x\) and vice versa. This is not true for even power functions. From Fig \(1.4\) we see that all power functions go through the point \((0,0)\). Even power functions have a local minimum at the origin whereas odd power functions do not.

Mastered Material Check

12. Highlight the \(y\)-axes and circle the origins in Fig 1.4.
13. Consider Figure 1.4: where do even power functions intersect? Odd?
14. Show that \(f(a)=a^{5}-3 a\) is an odd function.
15. Give an example of a function which is bounded.
16. Verify \(y=x^{2}\) is not one-to-one.
17. What graphical property do one-to-one functions share? Definition 1.1 (Local Minimum) A local minimum of a function \(f(x)\) is a point \(x_{\min }\) such that the value of \(f\) is larger at all sufficiently close points. Formally, \(f\left(x_{\min } \pm \varepsilon\right)>f\left(x_{\min }\right)\) for \(\varepsilon\) small enough.

Definition 1.1: Local Minimum

A local minimum of a function \(f(x)\) is a point \(x_{min}\) such that the value of \(f\) is larger at all sufficiently close points. Formally, \(f(x_{min} ±ε) > f(x_{min})\) for \(ε\) small enough.

Sketching a simple (two-term) polynomial

Based on our familiarity with power functions, we now discuss functions made up of such components. In particular, we extend the discussion to polynomials (sums of power functions) and rational functions (ratios of such functions). We also develop skills in sketching graphs of these functions.

Example 1.7 (Sketching a simple cubic polynomial)

Sketch a graph of the polynomial

\[y=p(x)=x^{3}+a x \]

How would the sketch change if the constant a changes from positive to negative?

Adjust the slider to see how positive and negative values of the coefficient a affect the shape of this simple polynomial.

Solution

The polynomial in Equation (\(\PageIndex{1}\)) has two terms, each one a power function. Let us consider their effects individually. Near the origin, for \(x \approx 0\) the term \(a x\) dominates so that, close to \(x=0\), the function behaves as

\[y \approx a x . \nonumber \]

This is a straight line with slope \(a\). Hence, near the origin, if \(a>0\) we would see a line with positive slope, whereas if \(a<0\) the slope of the line should be negative. Far away from the origin, the cubic term dominates, so

\[y \approx x^{3} \nonumber \]

at large (positive or negative) \(x\) values. Figure \(1.5\) illustrates these ideas.

In the first row we see the behavior of \(y=p(x)=x^{3}+a x\) for large \(x\), in the second for small \(x\). The last row shows the graph for an intermediate range. We might notice that for \(a<0\), the graph has a local minimum as well as a local maximum. Such an argument already leads to a fairly reasonable sketch of the function in Equation (\(\PageIndex{1}\)). We can add further details using algebra to find zeros - that is where \(y=p(x)=0\).

Example 1.8 (Zeros)

Find the places at which the polynomial Equation (\(\PageIndex{1}\)) crosses the \(x\) axis, that is, find the zeros of the function \(y=x^{3}+a x\).

Solution

The zeros of the polynomial can be found by setting

\[y=p(x)=0 \quad \Rightarrow \quad x^{3}+a x=0 \quad \Rightarrow \quad x^{3}=-a x . \nonumber \]

The above equation always has a solution \(x=0\), but if \(x \neq 0\), we can cancel and obtain

\[x^{2}=-a . \nonumber \]

Figure 1.5: The graph of the polynomial \(y=p(x)=x^{3}+a x\) can be obtained by combining its two power function components. The cubic "arms" \(y \approx x^{3}\) (top row) dominate for large \(x\) (far from the origin), while the linear part \(y \approx a x\) (middle row) dominates near the origin. When these are smoothly connected (bottom row) we obtain a sketch of the desired polynomial. Shown here are three possibilities, for \(a<0, a=0, a>0\), left to right. The value of \(a\) determines the slope of the curve near \(x=0\) and thus also affects presence of a local maximum and minimum (for \(a<0\) ).

This would have no solutions if \(a\) is a positive number, so that in that case, the graph crosses the \(x\) axis only once, at \(x=0\), as shown in Figure 1.5. If \(a\) is negative, then the minus signs cancel, so the equation can be written in the form

\[x^{2}=|a| \nonumber \]

and we would have two new zeros at

\[x=\pm \sqrt{|a|} . \nonumber \]

For example, if \(a=-1\) then the function \(y=x^{3}-x\) has zeros at \(x=0,1,-1\).

Mastered Material Check

17. Justify why the linear term dominates near the origin, while the cubic term dominates further out.
18. Sketch the graph of any function with horizontal asymptote \(y=2\).

Example 1.9 (A more general case)

Explain how you would use the ideas of Example \(1.7\) to sketch the polynomial \(y=p(x)=a x^{n}+b x^{m}\). Without loss of generality, you may assume that \(n>m \geq 1\) are integers.

Solution

As in Example 1.7, this polynomial has two terms that dominate at different ranges of the independent variable. Close to the origin, \(y \approx b x^{m}\) (since \(m\) is the lower power) whereas for large \(x, y \approx a x^{n}\). The full behavior is obtained by smoothly connecting these pieces of the graph. Finding zeros can refine the graph.

Mastered Material Check

19. Find the zeros of \(y=x^{3}+3 x\).

A step back. The reasoning used here is an important first step in sketching the graph of a polynomial. In the ensuing chapters, we develop specialized methods to find zeros of more complicated functions (using an approximation technique called Newton’s method). We also apply calculus tools to determine points at which the function attains local maxima or minima (called critical points), and how it behaves for very large positive or negative values of \(x\). That said, the elementary steps described here remain useful as a quick approach for visualizing the overall shape of a graph.

Sketching a simple rational function

We apply similar reasoning to consider the graphs of simple rational functions. A rational function is a function that can be written as

\[y=\frac{p_{1}(x)}{p_{2}(x)}, \quad \text { where } \quad p_{1}(x) \text { and } p_{2}(x) \text { are polynomials. } \nonumber \]

Example 1.10

(A rational function) Sketch the graph of the rational function

\[y=\frac{A x^{n}}{a^{n}+x^{n}}, \quad x \geq 0 \]

What properties of your sketch depend on the power \(n\) ? What would the graph look like for \(n=1,2,3\) ?

Adjust the sliders to see how the values of n, A, and a affect the shape of the rational function in (\(\PageIndex{2}\)).

Solution

We can break up the process of sketching this function into the following steps:

• The graph of the function in Equation (\(\PageIndex{2}\)) goes through the origin (at \(x=0\), we see that \(y=0\) ).
• For very small \(x\), (i.e., \(x<<a\) ) we can approximate the denominator by the constant term \(a^{n}+x^{n} \approx a^{n}\), since \(x^{n}\) is negligible by comparison, so that

\[y=\frac{A x^{n}}{a^{n}+x^{n}} \approx \frac{A x^{n}}{a^{n}}=\left(\frac{A}{a^{n}}\right) x^{n} \quad \text { for small } x . \nonumber \]

This means that near the origin, the graph looks like a power function, \(y \approx C x^{n}\left(\right.\) where \(\left.C=A / a^{n}\right)\).

• For large \(x\), i.e. \(x>>a\), we have \(a^{n}+x^{n} \approx x^{n}\) since \(x\) overtakes and dominates over the constant \(a\), so that

\[y=\frac{A x^{n}}{a^{n}+x^{n}} \approx \frac{A x^{n}}{x^{n}}=A \text { for large } x . \nonumber \]

This reveals that the graph has a horizontal asymptote \(y=A\) at large values of \(x\).

The results are displayed in Figure 1.6.

Figure 1.6: The rational functions Equations(\(\PageIndex{2}\)) with \(n = 1, 2, 3\) are compared on this graph. Close to the origin, the function behaves like a power function, whereas for large x there is a horizontal asymptote at \(y = A\). As n increases, the graph becomes flatter close to the origin, and steeper in its rise to the asymptote.

Mastered Material Check

20. Why is \(a^{n}\) a constant?
21. Sketch the graph of any function with horizontal asymptote \(y=2\). - Since the function behaves like a simple power function close to the origin, we conclude directly that the higher the value of \(n\), the flatter is its graph near 0 . Further, large \(n\) means sharper rise to the eventual asymptote.