Chapter SH, Problem 9.6EP

To determine

To find: The transformation in the function $g\left(x\right)=-x^2-4$ if the function is related to the graph of $f\left(x\right)=x^2$ .

Answer to Problem 9.6EP

The graph of the function $g\left(x\right)=-x^2-4$ is translated 4 units down relative to the graph of the function $f\left(x\right)=x^2$ .

Explanation of Solution

Given information:

The given function are $f\left(x\right)=x^2$ and $g\left(x\right)=-x^2-4$ .

Calculation:

Consider the function $f\left(x\right)=x^2$ and $g\left(x\right)=-x^2-4$ .

Compare the function $f\left(x\right)=x^2$ with the standard form of the quadratic equation $y=a{x}^2+bx+c$ .

  $a_1=1,b_1=0,c_1=0$

Since, the value of $a_1$ is positive. So, the graph is reflected across the x -axis and opens upward.

Compare the function $g\left(x\right)=-x^2-4$ with the standard form of the quadratic equation $y=a{x}^2+bx+c$ .

  $a_2=-1,b_2=0,c_2=-4$

Since, the value of $a_2$ is negative. So, the graph is reflected across the x -axis and opens downward.

From the above, it is clear that the value of $c_2$ is $-4$ while the value of $c_1$ is 0. So, the graph of the function $g\left(x\right)=-x^2-4$ is translated 4 units down relative to the graph of the function.

Draw the graph for both the functions.

  

From the above figure, it is clear that the graph of the function $g\left(x\right)=-x^2-4$ is translated 4 units down relative to the graph of the function.

Therefore, the graph of the function $g\left(x\right)=-x^2-4$ is translated 4 units down relative to the graph of the function $f\left(x\right)=x^2$ .