Chapter 1.2, Problem 7QR

To determine

Toexplain:The transformation ofthe graph of $f\left(x\right)=x^2$ to the graph of $g\left(x\right)={\left(x+2\right)}^2-3$ .

Answer to Problem 7QR

Shift the graph of $f\left(x\right)=x^2$ two units to the left and three units downwards to get the graph of $g\left(x\right)={\left(x+2\right)}^2-3$ .

Explanation of Solution

Given information:The functions are $f\left(x\right)=x^2$ and $g\left(x\right)={\left(x+2\right)}^2-3$ .

Consider a function $f\left(x\right)$ . Shift the graph of $f\left(x\right)$ to the left by $k$ units to draw the graph of $f\left(x+k\right)$ and shift $f\left(x\right)$ to downwards by $k$ units to draw the graph of $f\left(x\right)-k$ .

The given functions are $f\left(x\right)=x^2$ and $g\left(x\right)={\left(x+2\right)}^2-3$ . First draw the graph of $f\left(x\right)=x^2$ . Now, shift the graph of $f\left(x\right)=x^2$ two units to the left and, then 3 units downwards.

Therefore, shift the graph of $f\left(x\right)=x^2$ two units to the left of origin and three units downwards to get the graph of $g\left(x\right)={\left(x+2\right)}^2-3$ .