How do I tell an exponential function from a quadratic like y = x^2?

Look at where the variable sits. In an exponential function the variable is in the exponent and the base is a fixed number, as in y = 3 · 2^x. In a quadratic the variable is the base and the exponent is a fixed number, as in y = 3x^2. They behave completely differently: doubling x multiplies a quadratic's output by 4, but adding 1 to x multiplies an exponential's output by b.

Why can't the base b be negative or equal to 1?

If b were negative, outputs would flip sign at every whole-number input and fractional inputs like x = 1/2 would require square roots of negatives, so there would be no continuous real curve to graph. If b = 1, then b^x = 1 for every x and the function collapses to the horizontal line y = a, which is linear rather than exponential.

Does the graph of an exponential function ever cross the x-axis?

No. For y = a · b^x with a ≠ 0, there is no input that makes the output zero, because a positive base raised to any power is positive and multiplying by a nonzero a keeps it nonzero. The x-axis is a horizontal asymptote: the curve gets closer and closer to it on one side but never touches it, so the function has no x-intercept.

What is the fastest way to find b from a table?

When the x-values go up by 1 each time, divide any output by the one directly before it. That quotient is b. Always check at least two pairs — if the ratios do not match, the data is not exponential. If the x-values are not spaced one unit apart, first identify the ratio across the full gap and then take the appropriate root, or use points that are one unit apart.

ALG1-6.3

Exponential Functions & Their Graphs

Learn to identify, evaluate, and graph exponential functions y = a·bˣ: find a and b, tell growth from decay, and spot constant ratios versus constant differences.

What you'll do in this lesson

A voice-first session with the Crimsora tutor on Exponential Functions & Their Graphs, then targeted practice and FRQs — with the tutor adapting to where you get stuck.

Start Exponential Functions & Their Graphs free

What this lesson covers

Linear functions add the same amount over and over. Exponential functions multiply by the same amount over and over — and that small change in the rule makes an enormous change in the picture. A population that doubles, a car that loses a fixed percent of its value each year, a folded sheet of paper: all of these are exponential.

In this lesson you will learn to read an exponential function written as

y = a \cdot b^x

, pull the starting value

a

and the base

b

straight out of the equation, decide whether the function grows or decays, and sketch its graph including the horizontal asymptote. You will also learn the single fastest test for telling exponential data from linear data in a table: linear data has a constant difference, exponential data has a constant ratio.

The Form y = a·b^x and What Each Letter Does

An exponential function has the variable in the exponent:

y = a \cdot b^x

Here

a

is the initial value and

b

is the base, also called the growth or decay factor. Two restrictions come with the definition:

a \neq 0

, and

b > 0

with

b \neq 1

. We require

b > 0

because something like

(-4)^{1/2}

is not a real number, and we exclude

b = 1

because

y = a \cdot 1^x

is just the constant function

y = a

.

The value of

a

is easy to read off the graph: substitute

x = 0

and you get

y = a \cdot b^0 = a \cdot 1 = a

. So $a$ is always the $y$ -intercept, the point

(0, a)

.

The base

b

tells you what happens for each increase of 1 in

x

. Multiply by

b

every time

x

goes up by one step. In

y = 5 \cdot 3^x

, the outputs are

5, 15, 45, 135, \dots

— each is triple the one before.

A common mistake is misreading which number is which. In

y = 2^x

there is no visible

a

, but

a = 1

, so the graph passes through

(0,1)

. In

y = 3x^2

the variable is in the base, not the exponent, so that is a quadratic, not exponential. Ask yourself every time: where is the

x

? If it is upstairs in the exponent, the function is exponential.

Growth Versus Decay

Once

a

is positive, the base alone decides the shape.

Condition	Behavior	What the graph does (left to right)
$b > 1$	Exponential growth	Rises, slowly at first, then very steeply
$0 < b < 1$	Exponential decay	Falls steeply at first, then flattens toward zero
$b = 1$	Not exponential	Flat horizontal line $y = a$

For

y = 4 \cdot 2^x

the base is

2 > 1

, so outputs double: growth. For

y = 4 \cdot \left(\frac{1}{2}\right)^x

the base is

\frac{1}{2}

, between 0 and 1, so outputs halve: decay.

Decimals trip people up. In

y = 200 \cdot (0.85)^x

, the base

0.85

is less than 1, so this decays — each year the quantity keeps 85 percent of what it had, losing 15 percent. In

y = 200 \cdot (1.06)^x

, the base

1.06

is greater than 1, so this grows by 6 percent each step. The dividing line is always the number 1, not the number 0.

A second trap: a negative

a

flips the graph below the

x

-axis. In

y = -3 \cdot 2^x

the outputs are

-3, -6, -12, \dots

— the values are getting more negative, so the graph falls, even though

b > 1

. The function is still described as exponential growth in magnitude, but its graph is a reflection of

y = 3 \cdot 2^x

across the

x

-axis. Unless a problem says otherwise, assume

a > 0

Graphs: Asymptotes, Domain, and Range

Every graph of

y = a \cdot b^x

has the same skeleton. Make a small table, plot four or five points, and connect them with a smooth curve — never a series of straight segments.

Take

y = 3 \cdot 2^x

$x$	$-2$	$-1$	$0$	$1$	$2$	$3$
$y$	$0.75$	$1.5$	$3$	$6$	$12$	$24$

Notice the negative

x

-values do not produce negative

y

-values. Since

2^{-2} = \frac{1}{4}

, the outputs shrink toward zero but never reach it. The line

y = 0

(the

x

-axis) is a horizontal asymptote: the curve gets arbitrarily close to it and never touches or crosses it.

For a basic exponential with

a > 0

, the domain is all real numbers — you may put any

x

in the exponent — and the range is

y > 0

. There is no

x

-intercept, because

a \cdot b^x

can never equal zero when

a \neq 0

.

For a decay function such as

y = 3 \cdot \left(\frac{1}{2}\right)^x

, the same table runs the other direction:

12, 6, 3, 1.5, 0.75, 0.375

. The curve is steep on the left and flattens on the right, hugging the

x

-axis as

x

grows.

Where students go wrong: drawing the curve so it eventually touches the axis, or stopping the sketch at

x = 0

as though negative inputs are illegal. Extend the curve in both directions and let it approach the asymptote without landing on it.

Constant Difference Versus Constant Ratio

Given a table of values with evenly spaced

x

-values, you can decide linear versus exponential in about ten seconds.

Subtract consecutive

y

-values. If the differences are all the same, the data is linear, and that common difference is the slope. Divide consecutive

y

-values. If the quotients are all the same, the data is exponential, and that common ratio is

b

$x$	$0$	$1$	$2$	$3$
Table A: $y$	$6$	$10$	$14$	$18$
Table B: $y$	$6$	$12$	$24$	$48$

Table A: differences are

4, 4, 4

— linear,

y = 4x + 6

. Table B: ratios are

\frac{12}{6} = 2

\frac{24}{12} = 2

\frac{48}{24} = 2

— exponential,

y = 6 \cdot 2^x

.

Two cautions. First, the

x

-values must increase by the same step for this to work; if the table jumps

x = 0, 1, 3, 4

, compare only the pairs one unit apart or account for the gap. Second, checking one pair is not enough. In the table

3, 6, 9, 18

the first ratio is 2 but the second is

1.5

, so the data is neither linear nor exponential.

The big-picture idea, and the reason this matters later: a linear quantity changes by equal amounts over equal intervals, while an exponential quantity changes by equal factors over equal intervals. That is why exponential growth eventually outruns any linear growth, no matter how steep the line starts.

Writing an Equation and Evaluating It

To build

y = a \cdot b^x

from information, find

a

first, then

b

.

If you are given the value at

x = 0

, that value is

a

immediately. Then take any two consecutive outputs and divide the later by the earlier to get

b

. From the table

x = 0, 1, 2

with

y = 8, 20, 50

a = 8

and

b = \frac{20}{8} = 2.5

(check:

\frac{50}{20} = 2.5

). So

y = 8 \cdot (2.5)^x

.

Evaluating requires care with order of operations. In

y = 8 \cdot (2.5)^x

x = 3

, the exponent applies only to

2.5

: compute

2.5^3 = 15.625

first, then multiply by 8 to get

125

. Multiplying

8 \cdot 2.5 = 20

first and then cubing gives

8000

, which is wrong. Exponents come before multiplication, always.

Negative exponents show up constantly here. Evaluate

y = 12 \cdot 3^x

x = -2

3^{-2} = \frac{1}{9}

, so

y = \frac{12}{9} = \frac{4}{3}

. A negative exponent does not make the answer negative — it makes it a reciprocal, so the output stays positive.

One more useful fact:

\left(\frac{1}{2}\right)^x = 2^{-x}

. A decay function can always be rewritten as a growth base with a negative exponent, which is why the decay graph looks exactly like the growth graph reflected across the

y

-axis.

Key terms

Exponential function.: A function of the form $y = a \cdot b^x$ where the variable appears in the exponent, with $a \neq 0$ , $b > 0$ , and $b \neq 1$ .
Initial value ( $a$ ).: The output when $x = 0$ ; it is the $y$ -intercept of the graph, since $b^0 = 1$ .
Base / growth factor ( $b$ ).: The number the output is multiplied by each time $x$ increases by 1.
Exponential growth.: Behavior when $b > 1$ (with $a > 0$ ): outputs increase by a constant factor and the graph rises ever more steeply.
Exponential decay.: Behavior when $0 < b < 1$ (with $a > 0$ ): outputs shrink by a constant factor and the graph falls toward the $x$ -axis.
Horizontal asymptote.: A horizontal line the graph approaches without ever reaching; for $y = a \cdot b^x$ it is the line $y = 0$ .
Common ratio.: The constant quotient of consecutive outputs in a table with equally spaced inputs; it identifies exponential data and equals $b$ .
Common difference.: The constant amount added between consecutive outputs in a table with equally spaced inputs; it identifies linear data and equals the slope.

Worked example

A table lists the value of a piece of equipment over time: at

x = 0

years the value is 24,000 dollars; at

x = 1

it is 18,000 dollars; at

x = 2

it is 13,500 dollars; at

x = 3

it is 10,125 dollars. Show that the data is exponential rather than linear, write an equation of the form

y = a \cdot b^x

, classify it as growth or decay, and find the value after 5 years.

Step 1 — Test for a constant difference.

18000 - 24000 = -6000

, but

13500 - 18000 = -4500

. The differences are not equal, so the data is not linear.

Step 2 — Test for a constant ratio.

\frac{18000}{24000} = 0.75

\frac{13500}{18000} = 0.75

, and

\frac{10125}{13500} = 0.75

. The ratio is constant, so the data is exponential with

b = 0.75

.

Step 3 — Identify $a$ . The value at

x = 0

is 24,000, and

a

is always the output at

x = 0

, so

a = 24000

.

Step 4 — Write the equation.

y = 24000 \cdot (0.75)^x

.

Step 5 — Classify. Since

0 < 0.75 < 1

, this is exponential decay. Each year the equipment keeps 75 percent of its value, losing 25 percent.

Step 6 — Evaluate at $x = 5$ . Apply the exponent before multiplying:

0.75^5 = 0.2373046875

. Then

y = 24000 \cdot 0.2373046875 = 5695.3125

.

After 5 years the equipment is worth about 5,695 dollars. As a reasonableness check, the value should sit below the year-3 value of 10,125 dollars and stay positive — which it does, since a decay curve approaches zero without reaching it.

Practice questions

Which function represents exponential decay with a

y

-intercept of 7?

$y = 7 \cdot (1.2)^x$
$y = 7 \cdot (0.4)^x$
$y = 0.4 \cdot 7^x$
$y = 7x + 0.4$

Answer: $y = 7 \cdot (0.4)^x$

The

y

-intercept is

a

, the number in front, so

a

must be 7 — that rules out the third and fourth options (the third has

a = 0.4

, and the fourth is linear anyway). Decay requires

0 < b < 1

. In the first option

b = 1.2 > 1

, which is growth. Only

y = 7 \cdot (0.4)^x

has both

a = 7

and a base between 0 and 1.

Evaluate

f(x) = 5 \cdot 4^x

x = -2

and at

x = 3

$f(-2) = -80$ and $f(3) = 320$
$f(-2) = \frac{5}{16}$ and $f(3) = 320$
$f(-2) = \frac{5}{16}$ and $f(3) = 8000$
$f(-2) = -\frac{5}{16}$ and $f(3) = 60$

Answer: $f(-2) = \frac{5}{16}$ and $f(3) = 320$

For

x = -2

4^{-2} = \frac{1}{16}

, so

f(-2) = 5 \cdot \frac{1}{16} = \frac{5}{16}

. A negative exponent produces a reciprocal, not a negative output — the graph never dips below the

x

-axis when

a > 0

. For

x = 3

: apply the exponent first,

4^3 = 64

, then multiply,

5 \cdot 64 = 320

. Multiplying

5 \cdot 4 = 20

first and cubing gives 8000, which violates the order of operations.

A table shows

x = 0, 1, 2, 3

paired with

y = 150, 180, 216, 259.2

. Decide whether the relationship is linear or exponential, justify your answer with calculations, write the equation, and describe how the graph behaves as

x

increases.

Answer: Exponential, with equation $y = 150 \cdot (1.2)^x$ ; the graph rises from the intercept $(0,150)$ and gets steeper as $x$ increases.

Check differences first:

180 - 150 = 30

but

216 - 180 = 36

. Not constant, so it is not linear. Check ratios:

\frac{180}{150} = 1.2

\frac{216}{180} = 1.2

\frac{259.2}{216} = 1.2

. The ratio is constant, so the data is exponential with

b = 1.2

. The output at

x = 0

is 150, so

a = 150

and the equation is

y = 150 \cdot (1.2)^x

. Because

b = 1.2 > 1

, this is growth: each step multiplies the value by 1.2, a 20 percent increase. The curve passes through

(0,150)

, climbs at an increasing rate to the right, and to the left it decreases toward the horizontal asymptote

y = 0

without ever touching it.

FAQ

How do I tell an exponential function from a quadratic like $y = x^2$ ?: Look at where the variable sits. In an exponential function the variable is in the exponent and the base is a fixed number, as in $y = 3 \cdot 2^x$ . In a quadratic the variable is the base and the exponent is a fixed number, as in $y = 3x^2$ . They behave completely differently: doubling $x$ multiplies a quadratic's output by 4, but adding 1 to $x$ multiplies an exponential's output by $b$ .
Why can't the base $b$ be negative or equal to 1?: If $b$ were negative, outputs would flip sign at every whole-number input and fractional inputs like $x = \frac{1}{2}$ would require square roots of negatives, so there would be no continuous real curve to graph. If $b = 1$ , then $b^x = 1$ for every $x$ and the function collapses to the horizontal line $y = a$ , which is linear rather than exponential.
Does the graph of an exponential function ever cross the $x$ -axis?: No. For $y = a \cdot b^x$ with $a \neq 0$ , there is no input that makes the output zero, because a positive base raised to any power is positive and multiplying by a nonzero $a$ keeps it nonzero. The $x$ -axis is a horizontal asymptote: the curve gets closer and closer to it on one side but never touches it, so the function has no $x$ -intercept.
What is the fastest way to find $b$ from a table?: When the $x$ -values go up by 1 each time, divide any output by the one directly before it. That quotient is $b$ . Always check at least two pairs — if the ratios do not match, the data is not exponential. If the $x$ -values are not spaced one unit apart, first identify the ratio across the full gap and then take the appropriate root, or use points that are one unit apart.

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