Exponential Functions and Graphs SAT Notes

An exponential function has a variable in the exponent. In the source-supported form $$f(x)=ab^x$$ , the scale $$a$$ gives the output at $$x=0$$ , while the base $$b$$ gives the constant multiplicative change for each one-unit increase in $$x$$ .

Learning objectives

Identify $$a$$ , $$b$$ , and the vertical-axis intercept.
Classify growth when $$b>1$$ and decay when $$0<b<1$$ .
Build a value table and plot exact points from the same equation.
Compare exponential models using values, percent differences, and average rate of change.

Core exponential structure

Source-supported exponential form

\[f(x)=ab^x,\qquad a\ne0,\quad b>0,\quad b\ne1\]

The source uses these conditions. The positive-valued graph and range statements below additionally assume $$a>0$$ .

Growth versus decay

Exact curves generated from exact values

The first panel starts at 3 and multiplies by 1.5; the second starts at 12 and multiplies by one half. Each labeled point is calculated from its model.

Exponential growth

Exponential decay

Read the base before judging the curve

Growth

If $$b>1$$ , multiplying repeatedly makes outputs increase as $$x$$ increases.

Decay

If $$0<b<1$$ , multiplying repeatedly makes outputs decrease as $$x$$ increases.

Exact values behind two Chapter 12 example graphs
Input	Growth output	Decay output
$$0$$	$$3$$	$$12$$
$$1$$	$$4.5$$	$$6$$
$$2$$	$$6.75$$	$$3$$
$$3$$	$$10.125$$	$$1.5$$

Domain and range in the positive-scale case

Compare exponential models numerically

Two growth models with different starts and factors

Worked example

Different starts can be overtaken

Model A is $$18(1.5)^x$$ , and Model B is $$30(1.25)^x$$ . Compare them at $$x=0$$ and $$x=4$$ .

Initial values
At $$x=0$$ , Model A is $$18$$ and Model B is $$30$$ .
Later values
At $$x=4$$ , Model A is $$18(1.5)^4=91.125$$ , while Model B is $30(1.25)^4\approx73.242$ .
Interpret
Model A begins lower but its larger multiplier eventually produces the larger output.

Model B is larger initially; Model A is larger at

\(x=4\)

Exponential versus linear change

Look for differences or ratios

Linear

Equal input steps produce a constant additive difference.

Exponential

Equal input steps produce a constant multiplicative ratio or repeated percentage change.

Common mistakes and traps

Treating the exponent as a coefficient or a constant additive change.
Calling $$a$$ the growth factor or calling $$b$$ the initial value.
Classifying $$0<b<1$$ as growth.
Missing that $$f(0)=a$$ , not $$0$$ .
Using a straight line for repeated percentage decay.
Comparing visual steepness without checking axis scales.

Mini check

Read the parameters

For $$q(x)=40(0.7)^x$$ , which statement is true?

It starts at $$0.7$$ and grows.
It starts at $$40$$ and decays.
It starts at $$40$$ and grows.
It has a constant difference of $$-0.3$$ .

Show answer and explanation

Answer: It starts at $$40$$ and decays.

$$q(0)=40$$ , and the base $$0.7$$ lies between $$0$$ and $$1$$ .

Key takeaways

What to remember

In $$ab^x$$ , $$a$$ is the output at $$x=0$$ and $$b$$ is the one-step factor.
A base above $$1$$ gives growth; a base between $$0$$ and $$1$$ gives decay.
Generate graph points from the equation rather than guessing a curve.
Compare models using exact values and the stated axis scale.

Continue learning

Put these notes into practice

Apply the ideas with SAT-style questions, then reinforce key details with flashcards.

Go to Practice Review Flashcards

Input	Growth output	Decay output
$\(0\)$	$\(3\)$	$\(12\)$
$\(1\)$	$\(4.5\)$	$\(6\)$
$\(2\)$	$\(6.75\)$	$\(3\)$
$\(3\)$	$\(10.125\)$	$\(1.5\)$