Chapter 2.3, Problem 69AYU

73. $h\left(x\right)=x^2-2x$

(a) Find the average rate of change from 2 to 4.

(b) Find an equation of the secant line containing $\left(2,h\left(2\right)\right)$ and $\left(4,h\left(4\right)\right)$.

(c) Using a graphing utility, draw the graph of h and the secant line obtained in part (b) on the same screen.

To determine

To find: The following values using the function $h\left(x\right)=x^2-2x$.

a. Average rate of change of a function $h\left(x\right)$ from 2 to 4.

Answer to Problem 69AYU

a. Average rate of change of a function $h\left(x\right)$ from 2 to 4 is $-1$.

Explanation of Solution

Given:

The function $h\left(x\right)=x^2-2x$.

Calculation:

It is asked to find the average rate of change of $h\left(x\right)=x^2-2x$ the values from 2 to 4. Also the equation of the secant line containing $\left(2,h\left(2\right)\right)$ and $\left(4,h\left(4\right)\right)$.

a. The average rate of change of $h\left(x\right)$ from 2 to 4 is,

$\frac{\text{Δ}y}{\text{Δ}x}=\frac{h\left(4\right)-h\left(2\right)}{4-2}=\frac{8-0}{2}=4$

To determine

To find: The following values using the function $h\left(x\right)=x^2-2x$.

b. An equation of the secant line containing $\left(2,h\left(2\right)\right)$ and $\left(4,h\left(4\right)\right)$.

Answer to Problem 69AYU

b. An equation of the secant line containing $\left(2,h\left(2\right)\right)$ and $\left(4,h\left(4\right)\right)$ is $y=-x$.

Explanation of Solution

Given:

The function $h\left(x\right)=x^2-2x$.

Calculation:

It is asked to find the average rate of change of $h\left(x\right)=x^2-2x$ the values from 2 to 4. Also the equation of the secant line containing $\left(2,h\left(2\right)\right)$ and $\left(4,h\left(4\right)\right)$.

b. The line containing the two points $\left(a,f\left(a\right)\right)$ and $\left(b,f\left(b\right)\right)$ are called the secant line; its slope is $m_{sec}=\frac{f\left(b\right)-f\left(a\right)}{b-a}=\frac{f\left(a+h\right)-f\left(a\right)}{h}$.

An equation of the secant line containing $\left(2,h\left(2\right)\right)$ and $\left(4,h\left(4\right)\right)$ becomes,

$m_{sec}=\frac{h\left(4\right)-h\left(2\right)}{4-2}=4$

One of the coordinate is $\left(2,h\left(2\right)\right)=\left(2, 0\right)$.

Equation containing a point and slope is given by $y-y_1=m\left(x-x_1\right)$.

$y-0=4\left(x+2\right)$

$y=4x-8$

To determine

To find: The following values using the function $h\left(x\right)=x^2-2x$.

c. Draw the graph of $h$ and the secant line obtained in part (b) on the same screen using graphing utility.

Answer to Problem 69AYU

c.

Explanation of Solution

Given:

The function $h\left(x\right)=x^2-2x$.

Calculation:

It is asked to find the average rate of change of $h\left(x\right)=x^2-2x$ the values from 2 to 4. Also the equation of the secant line containing $\left(2,h\left(2\right)\right)$ and $\left(4,h\left(4\right)\right)$.

c. Graph: