Question

A function$f$is given, and the indicated transformations are applied to its graph (in the given order). Write an equation for the final transformed graph.

role="math" localid="1645440239035" $f\left(x\right)=x^2$; stretch vertically by a factor of 2,shift downward 2 units and shift 3 units to the right

Short answer

The transformed equation is$f\left(x\right)=2{(x-3)}^2-2$

Step-by-step solution

Step 1. Concept of vertical stretching and upward or downward and left or right shifting.

To graph$y=cf\left(x\right)$

If $c>1$;stretch the graph of$y=f\left(x\right)$ vertically by a factor of c

Adding a constant to a function shifts the graph upward or downward if the constant is positive or negative respectively.

To graph $y=f(x-c)$, shifts the graph$y=f\left(x\right)$ to the right c units

And, to graph$y=f(x+c)$ shifts the graph$y=f\left(x\right)$ to the left c units

Step 2. Given Problem.

Given that,

$f\left(x\right)=x^2$; Stretch vertically by a factor of 2,shift downward 2 units and shift 3 units to the right

Step 3. Determining the transformed equation.

Here, first stretch vertically by a factor of 2, then transformed equation is

$f\left(x\right)=2x^2$

And 2 is a constant which shifted downward, then

$f\left(x\right)=2x^2-2$

And 3 is a constant which shifted right, then

$f\left(x\right)=2{(x-3)}^2-2$