Question

Big Ideas:
Explain your reasoning.

Answer 1

Stretch the graph of f(x) = x + 3 vertically by a factor of 2.

The "same" transformations result in f(x) = 2x + 5

The "different" transformation results in f(x) = 2x + 6

Explanation:

Transformation Rules

`f(x)+a \implies f(x) \: \textsf{translated $a$ units up}`

`f(x+a) \implies f(x) \: \textsf{translated $a$ units left}`

`a\:f(x) \implies f(x) \: \textsf{stretched parallel to the $y$-axis (vertically) by a factor of $a$}`

`f(ax) \implies f(x) \: \textsf{stretched parallel to the $x$-axis (horizontally) by a factor of $(1)/(a)$}`

Carry out the given transformations.

Given function:

`f(x) = 2x + 3`

Translation of 2 units up:

`\begin{aligned}\implies f(x) + 2&= 2x + 3 + 2\\&=2x+5\end{aligned}`

Given function:

`f(x) =x + 3`

Vertical stretch by a factor of 2:

`\begin{aligned}\implies 2f(x)&=2(x+3)\\&=2x+6\end{aligned}`

Given function:

`f(x) = x + 5`

Horizontal shrink by a factor of 1/2:

`\implies f\left((1)/((1)/(2))x\right)=f(2x)=2x+5`

Given function:

`f(x) = 2x + 3`

Translation of 1 unit left:

`\begin{aligned}\implies f(x+1) &= 2(x+1) + 3\\&=2x+5\end{aligned}`

Three of the transformations result in the same function f(x) = 2 + 5.

Therefore, the transformation that does not belong with the other three is:

• Stretch the graph of f(x) = x + 3 vertically by a factor of 2.