Question

the function below shows the portion of the even function f(x) for x≥0. sketch the portion of f(x) for x<0. the graph will be graded. show all work on the worksheet. calculate the average rate of change of f(x) over the interval -7≤x≤5. express in simplest form. given the entire graph of f(x), how many solutions are there to f(x)=4? justify.

Explanation:

Step1: Recall property of even function

For an even function \(f(x)=f( - x)\). So the graph for \(x < 0\) is the reflection of the graph for \(x\geq0\) across the \(y\) - axis.

Step2: Recall average - rate - of - change formula

The average rate of change of a function \(y = f(x)\) over the interval \([a,b]\) is \(\frac{f(b)-f(a)}{b - a}\). Here \(a=-7\) and \(b = 5\). First, since \(f(x)\) is even, \(f(-7)=f(7)\). From the graph, we need to estimate \(f(5)\) and \(f(-7)=f(7)\). Let's assume we can read the values of the function from the graph: \(f(5)=y_1\) and \(f(-7)=f(7)=y_2\). Then the average rate of change is \(\frac{f(5)-f(-7)}{5-(-7)}=\frac{y_1 - y_2}{12}\).

Step3: Find solutions of \(f(x)=4\)

To find the number of solutions of \(f(x)=4\), we draw the horizontal line \(y = 4\) on the entire graph (after sketching the part for \(x<0\) using the even - function property). Count the number of intersection points of \(y = 4\) and \(y = f(x)\).

Answer:

For sketching: Reflect the \(x\geq0\) part of the graph across the \(y\) - axis. For average rate of change: Calculate \(\frac{f(5)-f(-7)}{12}\) after reading \(f(5)\) and \(f(-7)=f(7)\) from the graph. For number of solutions of \(f(x)=4\): Count the intersection points of \(y = 4\) and the entire graph of \(y = f(x)\).