Chapter 1.7, Problem 66E

To determine

To find:if the given statement is true or false.

Answer to Problem 66E

False

Explanation of Solution

Given:

The statement: “If the graph of the parent function $f\left(x\right)=x^2$ is shifted six units to the right, three units up, and reflected in the x -axis, then the point $\left(-2,19\right)$ will lie on the graph of the transformation.”

Consider the function $f\left(x\right)=x^2$ .

Shifting the function 6 units to the left, the new function is:

  $g\left(x\right)={\left(x-6\right)}^2$

Now, shifting the above function 3 units up, the new function is:

  $h\left(x\right)={\left(x-6\right)}^2+3$

Lastly reflecting the above function about the x -axis, the new function is:

  $\begin{array}{c}k\left(x\right)=-h\left(x\right)\\ =-\left[{\left(x-6\right)}^2+3\right]\\ =-{\left(x-6\right)}^2-3\end{array}$

Now, the objective is to verify if the point $\left(-2,19\right)$ lie on the graph of this function or not.

Substitute $x=-2$ in the function $k\left(x\right)=-{\left(x-6\right)}^2-3$ ,

  $\begin{array}{c}k\left(-2\right)=-{\left(-2-6\right)}^2-3\\ =-{\left(-8\right)}^2-3\\ =-64-3\\ =-67\end{array}$

Clearly, $k\left(-2\right)\ne 19$

So, the point $\left(-2,19\right)$ does not lie on the graph of the transformation.

Conclusion:

The statement is false.