Algebra 2 Lesson

Transformations of Parent Functions: Discovering a, b, h, k

Shared by A STEM Educator

11th GradeBTC Style / Thinking Classrooms90 min

Learning Objective

Students will analyze and apply the parameters a, b, h, and k in f(x) = a·g(b(x−h))+k to describe, predict, and produce transformations of parent functions including quadratic, absolute value, square root, and reciprocal families.

Lesson Overview

Students use a discovery-based investigation across multiple parent function families to isolate the effects of each parameter in f(x) = a·g(b(x−h))+k. Tasks thin-slice from single vertical shifts to multi-parameter transformations, culminating in coordinate mapping and error analysis across function families.

Materials

Vertical whiteboards or chart paper
Dry-erase markers (multiple colors)
Pencils and paper for note sheet
Graphing calculator or phone calculator (Desmos app)

Scaffolded Task Progression

1Task 1

The parent function is

g (x) = x^{2}

. Your group's function is

f (x) = x^{2} + 3

.

1. Make a table of values for

g (x)

using

x = - 2, - 1, 0, 1, 2

.
2. Make a table of values for

f (x)

using the same

x

-values.
3. What changed between corresponding

y

-values? Write one sentence describing what happened to the graph.

2Task 2

Same parent function

g (x) = x^{2}

. Now consider

f (x) = (x - 3)^{2}

.

1. Make a table for

f (x)

using

x = 1, 2, 3, 4, 5

.
2. Compare to your

g (x)

table from Task 1. What changed?
3. The vertex of

g (x)

is at

(0, 0)

. Where is the vertex of

f (x)

? Write a rule: 'When I replace

x

with

(x - h)

, the graph moves ___.'

3Task 3

Same parent

g (x) = x^{2}

. Compare these two functions by making tables (

x = - 2, - 1, 0, 1, 2

):

-

f (x) = 2 x^{2}

p (x) = \frac{1}{2} x^{2}

For each function: How did the

y

-values change compared to

g (x)

? Is the graph narrower or wider? Write a rule for what the

a

-value does.

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