Using Multiple Transformations to Graph Quadratic Functions

Step 1 of 4

Warm Up

Real-world idea

Who is right?

Alex and Bea both graph g(x) = −2(x + 1)² + 4 starting from y = x².

Alex: left 1 → stretch ×2 → reflect → up 4.

Bea: left 1 → up 4 → stretch ×2 → reflect.

They used the SAME four transformations. How could they end with different graphs?

Alex's final graph

y(x) = −2(x + 1)² + 4

Bea's final graph

y(x) = −2(x + 1)² − 8

Warm Up

Alex and Bea both start with y = x² and try to graph g(x) = −2(x + 1)² + 4.

Vertex form says the vertex should be (h, k). Whose final graph has the correct vertex?

aAlex — vertex (−1, 4)bBea — vertex (−1, −8)cBothdNeither

Keep Bea in mind. By the end of Learn you'll be able to point to the exact step where her reasoning went wrong.

Step 2 of 4

Learn

Part 1 — Break the equation apart

Read g(x) = −2(x + 1)² + 4 the way it is calculated: work on x inside the bracket, square, multiply by a, then add k.

Building −2(x + 1)² + 4 one stage at a time

y = x²parent
y = (x + 1)²horizontal translation: left 1
y = 2(x + 1)²vertical stretch ×2
y = −2(x + 1)²reflection in the x-axis
y = −2(x + 1)² + 4vertical translation: up 4

Part 2 — Why this order?

The order follows the operations in the equation.

• Inside the bracket happens first → horizontal translation.

• Then the output is multiplied by a → stretch/compression and reflection.

• Last, k is added → vertical translation.

Why k goes last

A stretch multiplies EVERY y-value. If you shift up 4 first, the vertex is at y = 4 — then stretching ×2 drags it to y = 8, and reflecting sends it to −8. That's exactly what happened to Bea.

Stretch and reflection both just multiply y-values, so they can be done in either order — or together, as one multiplication by a.

Try it

Which list describes all the transformations in y = 3(x − 2)² − 5?

aRight 2, vertical stretch by 3, down 5bLeft 2, vertical stretch by 3, down 5cRight 2, vertical compression by 3, down 5dRight 2, reflection, down 5

Try it

Which is an appropriate order for graphing y = −0.5(x + 3)² + 2?

aUp 2, left 3, compress ½, reflectbLeft 3, compress by ½, reflect in the x-axis, up 2cReflect, up 2, left 3, compress ½dCompress ½, up 2, reflect, left 3

Try it

Start with the point (2, 4) on y = x². Apply: shift left 3, then vertical compression by ½. What is the point's new y-coordinate?

Your answer

Part 3 — Transformation Challenge

Build each target from y = x² using transformation cards. Drag cards into your sequence (or tap to add, and use ↑ ↓ to reorder).

Press Apply to watch the graph change stage by stage. If it doesn't match, you'll see the stage where it went off track.

Transformation Challenge — Round 1 of 4 · Two transformations

Build g(x) = (x − 2)² − 3 from y = x². Drag cards into the sequence, or tap a card to add it.

y = x²
target
your graph

Cards

Your sequence

Drop or tap cards here.

Worked example

Graph y = 0.5(x − 4)² − 3 using multiple transformations. Track the vertex and the point (2, 4).

Step 1
Right 4.
(0, 0) → (4, 0); (2, 4) → (6, 4)
Step 2
Vertical compression ×0.5.
(4, 0) → (4, 0); (6, 4) → (6, 2)
Step 3
Down 3.
(4, 0) → (4, −3); (6, 2) → (6, −1)
Check
Substitute x = 6.
0.5(2)² − 3 = −1 ✓

Answer: Vertex (4, −3); (6, −1) and its mirror (2, −1).

Using Multiple Transformations to Graph Quadratic Functions

Warm Up

Who is right?

Learn

Part 1 — Break the equation apart

Part 2 — Why this order?

Part 3 — Transformation Challenge

Practice

Build it

Apply it

Challenge

Explain it

Mastery Check

Lesson 1.6 mastery check