A.4(A)·Algebra II·Function TransformationsSupporting

Parent Functions

Identify and sketch graphs of parent functions, including exponential functions.

Students identify an exponential parent function by removing transformations and checking its intercept, direction, and horizontal asymptote.

Lesson plan and worked examples by TEKSForge · not a TEA-issued lesson. Videos by independent educators.

Archived course: TEKS code and readiness label from TEA's 2013 Algebra II key. Algebra II is not one of Texas's current required STAAR EOC assessments, and this lesson does not change the Biology and Algebra I Study Plans.

Practice the released item Ask the tutor All Algebra II released practice50 minutes

Learn the concept

Remove transformations

Within a fixed-base exponential family, the parent of $y=a b^{x-h}+k$ is $y=b^x$ . Remove shifts and the outside multiplier, not the given base.

Build defining points

For $b>1$ , the points $(-1,1/b)$ , $(0,1)$ , and $(1,b)$ lie on the parent. Outputs are positive and increase from left to right.

Use end behavior

For $b>1$ , $b^x$ approaches zero as x decreases without bound. Its horizontal asymptote is $y=0$ , domain is all real numbers, and range is $y>0$ . A downward shift changes the asymptote; reflection across the y-axis reverses the direction.

Worked example

Identify and sketch the parent for $p(x)=6^{x+4}-2$ .

Remove the horizontal and vertical shifts to obtain $y=6^x$ .
Plot $(-1,\frac16)$ , $(0,1)$ , and $(1,6)$ .
Draw an increasing curve above the x-axis that approaches $y=0$ on the left.

The parent is $y=6^x$ , increasing through $(0,1)$ with asymptote $y=0$ .

Watch the lesson

Texas public-school, Texas educator-created, and TEKS-aligned lessons, hosted by their creators. Each card names its source.

Texas public school video

Algebra 2 Identifying Parent Functions

UT IPSI · Kid2Kid / Texas Education Agency

Recognize exponential parent graphs by their shape, intercept, and asymptote.

Watch on YouTube

Try it yourself

TEKSForge practice · unscored

1. What is the parent of $y=3\cdot8^{x-1}+5$ ?
Show answer and reasoning
$y=8^x$
Remove the multiplier 3, the right shift 1, and the upward shift 5.
2. Which describes $y=5^x$ : increasing through $(0,1)$ with asymptote $y=0$ , or increasing through $(0,0)$ with asymptote $y=-1$ ?
Show answer and reasoning
The first description.
$5^0=1$ and positive powers approach zero for very negative inputs; the second description is shifted down.

Released STAAR practice

Texas Education Agency · archived 2013 released EOC items · scored with TEA's answer key; step-by-step explanations written by TEKSForge because this release has no rationales. Original graphs, tables, and diagrams are shown where the released item includes one.

Archived released STAAR Algebra II EOC · 2013 · Item 6 · 1 point

Which graph best represents the parent function of $y = 4^{(x+3)}$ ?

Graph F
Graph G
Graph H
Graph J

Show STAAR answer and explanation

Graph F

The parent function of $y=4^{(x+3)}$ is $y=4^x$ . It passes through $(0,1)$ , increases from left to right, and approaches the x-axis ( $y=0$ ) as x decreases. Graph F shows exactly those features.

Graph G: Graph G passes through the origin and levels off at $y=-1$ , so it is the parent function shifted down 1 unit.

Graph H: Graph H decreases from left to right through $(0,1)$ ; it shows $y=4^{-x}$ , the reflection of the parent across the y-axis.

Graph J: Graph J decreases and passes through the origin, so it is reflected and shifted down rather than the parent function.

Ask the tutor about this question

Start scored STAAR practice

Learning objectives

By the end of the lesson, students can:

Identify an exponential parent rule from a transformed expression.
Sketch key points of an increasing exponential parent.
Distinguish parents from shifted or reflected graphs.

Instructional sequence

1
Engage
5 min
Sort familiar parent-function shapes.
2
Direct instruction
10 min
Remove transformations while retaining the exponential base.
3
Guided practice
13 min
Build tables and sketch an exponential parent.
4
Compare and verify
15 min
Reject distractor graphs using intercepts, direction, and asymptotes.
5
Closure
7 min
Justify the selected parent on the exit ticket.

Misconceptions to address

“The parent must pass through the origin.”

Teacher move: An exponential parent has $b^0=1$ , so its y-intercept is 1.

“The added constant in the exponent stays in the parent.”

Teacher move: That constant translates the graph; the unshifted parent uses exponent x.

Materials

•Parent-function graph cards
•Coordinate grids
•Calculator

Teaching tips

Use three independent graph features.
Include a negative input in each table.
Keep the base fixed when removing a translation.

Exit ticket · 3–5 min

Identify the parent of $y=7^{x-2}$ and state its y-intercept and asymptote.

Show expected answer

$y=7^x$ ; y-intercept $(0,1)$ ; asymptote $y=0$ .

Sources & alignment

TEA 2013 Algebra II released test TEA 2013 Algebra II answer key and standards Texas Gateway, home of the UT IPSI Kid2Kid lessons

Alignment follows the standard TEA assigned to each 2013 released item. Video playback depends on YouTube access; written instruction remains available here.

Learn the concept

Remove transformations

Build defining points

Use end behavior

Worked example

Watch the lesson

Algebra 2 Identifying Parent Functions

Try it yourself

1. What is the parent of y=3⋅8x−1+5y=3\cdot8^{x-1}+5y=3⋅8x−1+5?

2. Which describes y=5xy=5^xy=5x: increasing through (0,1)(0,1)(0,1) with asymptote y=0y=0y=0, or increasing through (0,0)(0,0)(0,0) with asymptote y=−1y=-1y=−1?

Released STAAR practice

Archived released STAAR Algebra II EOC · 2013 · Item 6 · 1 point

Learning objectives

Instructional sequence

Engage

Direct instruction

Guided practice

Compare and verify

Closure

Misconceptions to address

1. What is the parent of $y=3\cdot8^{x-1}+5$ ?

2. Which describes $y=5^x$ : increasing through $(0,1)$ with asymptote $y=0$ , or increasing through $(0,0)$ with asymptote $y=-1$ ?