Chapter 12.6, Problem 42STP

To determine

To describe the translation of the graph of $f\left(x\right)=x^2$ to the graph of $g\left(x\right)={\left(x+4\right)}^2-3$.

Answer to Problem 42STP

Translated 4 units horizontally to the left and translated down 3 units vertically down.

Explanation of Solution

Given:

  $g\left(x\right)={\left(x+4\right)}^2-3$

Concept used:

The general equation of $f\left(x\right)=x^2$ is $f\left(x\right)={\left(x-\mathrm{h}\right)}^2+\mathrm{k}$.

Calculation:

Comparing the function $f\left(x\right)={\left(x-\mathrm{h}\right)}^2+\mathrm{k}$ with the general equation, the values are

  $h=-4,k=-3$ .

So, the graph of $f\left(x\right)=x^2$ is translated 4 units horizontally to the left and translated down 3 units vertically down to get the graph of $g\left(x\right)={\left(x+4\right)}^2-3$

Conclusion:

Therefore, the graph of $f\left(x\right)=x^2$ is translated 4 units horizontally to the left and translated down 3 units vertically down.