1.1: Power Functions

Learning Objectives

1. Interpret the shapes of power functions relative to one another.
2. Justify that power functions with low powers dominate near the origin, and power functions with high powers dominate far away from the origin.
3. Identify the points of intersection of two power functions.

Let us consider the power functions, that is functions of the form

\[y=f(x)=x^{n}, \nonumber \]

where \(n\) is a positive integer. Power functions are among the most elementary and "elegant" functions - we only need multiplications to compute their value at any point. They are thus easy to calculate, very predictable and smooth, and, from the point of view of calculus, very easy to handle.

Click on this link and then adjust the slider on this interactive desmos graph to see how the power n affects the shape of a power function in the first quadrant.

Figure 1.1: Graphs of a few power functions \(y=x^{n}\). All intersect at \(x=0,1\). As the power \(n\) increases, the graphs become flatter close to the origin, \((0,0)\), and steeper at large \(x\)-values.

From Figure 1.1, we see that the power functions \(\left(y=x^{n}\right.\) for powers \(n=\) \(2, \ldots, 5)\) intersect at \(x=0\) and \(x=1\). This is true for all positive integer powers. The same figure also demonstrates another fact helpful for curve-sketching: the greater the power \(n\), the flatter the graph near the origin and the steeper

Mastered Material Check

1. Can you define function?
2. Give an example of a polynomial function; a rational function. Wlick on this link and then adjust the slider on this interactive desmos graph to see how the power \(n\) affects the shape of a power function in the first quadrant. the graph beyond \(x>1\). This can be restated in terms of the relative size of the power functions. We say that close to the origin, the functions with lower powers dominate, while far from the origin, the higher powers dominate.

More generally, a power function has the form

\[y=f(x)=K \cdot x^{n} \nonumber \]

where \(n\) is a positive integer and \(K\), sometimes called the coefficient, is a constant. So far, we have compared power functions whose coefficient is \(K=1\). We can extend our discussion to a more general case as well.

Example 1.1

Find points of intersection and compare the sizes of the two power functions

\[y_{1}=a x^{n}, \quad \text { and } \quad y_{2}=b x^{m} . \nonumber \]

where \(a\) and \(b\) are constants. You may assume that both \(a\) and \(b\) are positive.

Solution

This comparison is a slight generalization of the previous discussion. First, we note that the coefficients \(a\) and \(b\) merely scale the vertical behavior (i.e. stretch the graph along the \(y\) axis). It is still true that the two functions intersect at \(x=0\); further, as before, the higher the power, the flatter the graph close to \(x=0\), and the steeper for large positive or negative values of \(x\). However, now another point of intersection of the graphs occur when

\[a x^{n}=b x^{m} \quad \Rightarrow \quad x^{n-m}=(b / a) . \nonumber \]

We can solve this further to obtain a solution in the first quadrant

\[x=(b / a)^{1 /(n-m)} . \nonumber \]

This is shown in Figure 1.2 for the specific example of \(y_{1}=5 x^{2}, y_{2}=2 x^{3}\).

Close to the origin, the quadratic power function has a larger value, whereas for large \(x\), the cubic function has larger values. The functions intersect when \(5 x^{2}=2 x^{3}\), which holds for \(x=0\) or \(x=\frac{5}{2}=2.5\).

If \(b / a\) is positive, then in general the value given in (1.1) is a real number.

Example 1.2

Determine points of intersection for the following pairs of functions:

1. \(y_{1}=3 x^{4}\) and \(y_{2}=27 x^{2}\),
2. \(y_{1}=\left(\frac{4}{3}\right) \pi x^{3}\) and \(y_{2}=4 \pi x^{2}\).

Figure 1.2: Graphs of two power functions, \(y=5 x^{2}\) and \(y=2 x^{3}\). Note that in many cases, the points of intersection are irrational numbers whose decimal approximations can only be obtained by a scientific calculator or by some approximation method (such as Newton’s Method, studied in Section 5.4)

Solution

(a) Intersections occur at \(x=0\) and at \(\pm(27 / 3)^{1 /(4-2)}=\pm \sqrt{9}=\pm 3\).

(b) These functions intersect at \(x=0,3\) but there are no other intersections at negative values of \(x\).

Mastered Material Check

3. Use Figure \(1.1\) to approximate when \(x^{5}=2\).
4. What is the first quadrant?

With only these observations we can examine a biological problem related to the size of cells. By applying these ideas, we can gain insight into why cells have a size limitation, as discussed in the next section.