Chapter 0.2, Problem 7QR

To determine

To find: Describe how the graph of $f(x)=x^2$ can be transform to the graph of $g(x)={\left(x+2\right)}^2-3$ .

Answer to Problem 7QR

The graph of $f(x)=x^2$ can be transformed into $g(x)={\left(x+2\right)}^2-3$ by shifting 3 units downwards vertically and 2 units left horizontally.

Explanation of Solution

Given information:

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

Calculation:

To find the transformation, compare the function to the parent function and check to see if there is a horizontal or vertical shift, reflection about the x-axis or y-axis, and if there is a vertical stretch.

Adding a constant to a function will shift its graph vertically i.e. $y=f(x)+c$ Adding

a positive constant c will shift the graph upward c units, while adding a negative constant

c will shift it downward c units.

If -3 added to $x^2$ it becomes $x^2-3$ and the graph will shift 3 units downwards.

A horizontal shift is represented in either the form $y=f\left(x-c\right)$ or $y=f\left(x+c\right)$ . In the graph of $y=f(x)$ . The value of $f(x-c)$ at $x$ is the same as the value of

  $f(x)$ at $\left(x-c\right)$ . Since $\left(x-c\right)$ is $c$ units to the left of $x$ , it follows that the graph of $y=f\left(x-c\right)$ is just the graph of $y=f(x)$ shifted to the right $c$ units. Likewise, the graph of $y=f\left(x+c\right)$ is the graph of $y=f(x)$ shifted to the left $c$ unit.

In the function $f(x)=x^2$ if 2 is added it becomes $f(x)={\left(x+2\right)}^2$ .The graph of $f(x)=x^2$ is shifted to horizontally 2 unit to the left.

Hence the graph of $f(x)=x^2$ can be transformed into $g(x)={\left(x+2\right)}^2-3$ by shifting 3 units downwards vertically and 2 units left horizontally.