Graphing Quadratic Functions Using Transformations

Step 1 of 4

Warm Up

Real-world idea

What do we already know?

Here is g(x) = 2(x − 1)² − 3. Before drawing anything, what can you read straight from the equation?

Warm Up

Without graphing, what can you say about g(x) = 2(x − 1)² − 3?

aVertex (−1, −3), opens up, widerbVertex (1, −3), opens up, narrowercVertex (1, 3), opens down, narrowerdVertex (2, −3), opens up, narrower

Warm Up

True or false: knowing the vertex, axis and direction of opening is enough to sketch the parabola accurately.

TrueFalse

Vertex, axis and opening give the skeleton. To get the width right, we need a few more points — and y = x² can give them to us.

Step 2 of 4

Learn

Part 1 — Useful parent points

These five points of y = x² are easy to remember: one step from the vertex rises 1, two steps rises 4.

Parent points

y(x) = x²

Part 2 — Moving the points

For g(x) = a(x − h)² + k, every parent point moves the same way:

• its x-value shifts by h (sideways)

• its y-value is multiplied by a, then shifted by k.

(x, y) on y = x² → (x + h, a·y + k)

Multiply by a first, then add k.

y = 0.5(x + 3)² − 4 (a = 0.5, h = −3, k = −4)

(0, 0) → (−3, −4)the vertex
(1, 1) → (1 − 3, 0.5·1 − 4) = (−2, −3.5)
(2, 4) → (2 − 3, 0.5·4 − 4) = (−1, −2)
mirror across x = −3: (−4, −3.5), (−5, −2)symmetry

Plot the moved points, then join them smoothly

y(x) = 0.5(x + 3)² − 4

y = x² (parent)

Try it

For g(x) = 3(x − 1)² + 2, the parent point (1, 1) moves to (2, ?). What is the y-coordinate?

Your answer

Try it

Under g(x) = −(x + 2)² + 1, where does the parent point (2, 4) go?

a(4, −3)b(0, −3)c(0, 5)d(−4, −3)

Use symmetry

You only need to transform points on one side. Every point has a mirror twin the same distance from the axis x = h.

Try it

A parabola has axis of symmetry x = 1 and passes through (4, 6). Which other point must be on it?

a(−4, 6)b(−2, 6)c(1, 6)d(4, −6)

Part 3 — Build the Parabola

Your turn to build. Place the vertex, choose the opening and the stretch, then move the two transformed points into place. Press Check — the feedback tells you which feature to fix.

Build the Parabola — Round 1 of 4 · Translations only

Target: g(x) = (x + 3)² − 2

y = x²
your parabola

Hollow points are the mirror images across the axis of symmetry.

Step 1 · Place the vertex (h, k)
vertex x0
vertex y0
Step 2 · Opening direction
Step 3 · Vertical stretch or compression |a|
Step 4 · Place transformed points
Move each point up or down to where (1, 1) and (2, 4) land after the transformation.
y at x=10
y at x=20

A repeatable graphing strategy

1. Read vertex form; identify a, h, k
2. Plot the vertex (h, k)
3. Draw the axis x = h
4. Opening: sign of a
5. Stretch/compression: |a|
6. Transform (1, 1) and (2, 4) → (1 + h, a + k), (2 + h, 4a + k)
7. Reflect those points across the axis
8. Draw a smooth curve, then check one point in the equation

Worked example

Graph y = −2(x − 2)² + 3.

Step 1
a = −2, h = 2, k = 3. Vertex (2, 3), axis x = 2.
Step 2
Opens down, vertical stretch by 2.
Step 3
(1, 1) → (3, −2·1 + 3) = (3, 1).
(2, 4) → (4, −8 + 3) = (4, −5)
Step 4
Mirror: (1, 1) and (0, −5).
Step 5
Check x = 4 in the equation.
−2(2)² + 3 = −5 ✓

Answer: Points (0, −5), (1, 1), (2, 3), (3, 1), (4, −5) joined smoothly.

Graphing Quadratic Functions Using Transformations

Warm Up

What do we already know?

Learn

Part 1 — Useful parent points

Part 2 — Moving the points

Part 3 — Build the Parabola

Practice

Build it

Apply it

Challenge

Explain it

Mastery Check

Lesson 1.5 mastery check