AFDA - Unit 1, Day 4: Dilations & Combining

Unit 1, Day 4: Dilations & Combining

October 30, 2025

Warm-Up: Review

Describe all the transformations for the equation below.

$f(x) = -(x+4)^2 - 2$

Hint: There are three transformations. What does the front negative do? The $(x+4)$? The $-2$ at the end?

I Do: Dilations (Stretching & Shrinking)

A dilation happens when we multiply the whole function by a number in front, which we call '$a$'.

$a \cdot f(x)$

• If $a > 1$, it's a vertical stretch (the graph gets skinnier).
  Example: $f(x)=3x^2$ is skinnier than the parent.
• If $0 < a < 1$, it's a vertical compression (the graph gets wider).
  Example: $f(x)=\frac{1}{3}x^2$ is wider than the parent.
• See it: In Desmos, type $y=x^2$, then $y=3x^2$, and $y=(1/3)x^2$.

We Do: Putting It All Together

This is the final form! It combines all four possible transformations.

$f(x) = a(x-h)^2 + k$

Let's describe $f(x) = -2(x-3)^2+5$ together:

1. Look inside parentheses: $(x-3)$ means it moves right 3.
2. Look at the front number: The negative on the $-2$ means it reflects over the x-axis.
3. Look at the front number again: The $2$ itself means it has a vertical stretch (gets skinnier).
4. Look at the end number: The $+5$ means it moves up 5.

Let's check: Graph $y=x^2$ and $y=-2(x-3)^2+5$ in Desmos to confirm all the moves.

You Do: Describe Everything

Now it's your turn to describe all the transformations.

$f(x) = \frac{1}{2}(x+1)^2 - 4$

List all the transformations you see:

• What does the $\frac{1}{2}$ in front do?
• What does the $(x+1)$ inside do?
• What does the $-4$ at the end do?
• Check your work: Graph the parent $y=x^2$ and the equation above in Desmos to see if you were right.

Independent Practice

Describe all transformations for each equation.

Exit Ticket

On a separate piece of paper, answer the following.