Exponential Functions and Their Graphs

Understanding y = aˣ, exponential growth and decay, and transformations

SPEC 6.1 • EXPONENTIALS AND LOGARITHMS

What is an Exponential Function?

Definition

An exponential function is a function of the form:

y = a^{x}

where

a > 0

,

a \neq = 1

, and

x \in R

The variable

x

is in the exponent (power), which is why these are called exponential functions.

Key Distinction

Exponential:

y = 2^{x}

— variable is in the power

Power function:

y = x^{2}

— variable is in the base

Why a > 0?

If

a \leq 0

, we get undefined or complex values for some

x

. For example,

(- 2)^{0.5} = - 2

is not real.

Graph of y = 2ˣ

Let's build the graph by calculating key points:

x

-3-2-10123

y = 2^{x}

\frac{1}{8}

\frac{1}{4}

\frac{1}{2}

1248

Key Features of y = 2ˣ

y-intercept: (0, 1) — because

2^{0} = 1

Horizontal asymptote:

y = 0

(the x-axis)

Always positive:

y > 0

for all

x

Increasing function: as

x

increases,

y

increases

Comparing y = 2ˣ and y = 3ˣ

Common Features

Both pass through (0, 1)

Both have asymptote

y = 0

Both are always positive

Both are increasing

Key Difference

y = 3^{x}

grows faster than

y = 2^{x}

Larger base = steeper curve

For

x > 0

:

3^{x} > 2^{x}

For

x < 0

:

3^{x} < 2^{x}

General Rule

For

a > b > 1

: the graph of

y = a^{x}

is steeper than

y = b^{x}

for positive

x

.

Exponential Growth (a > 1)

When the base

a > 1

, the function shows exponential growth.

Exponential Growth

For

y = a^{x}

where

a > 1

:

As

x \to + \infty

,

y \to + \infty

(grows without bound)

As

x \to - \infty

,

y \to 0

(approaches but never reaches 0)

Key Properties for a > 1

PropertyValuey-intercept(0, 1)Horizontal asymptote

y = 0

Domain

x \in R

Range

y > 0

BehaviourIncreasing (growth)

Real-World Examples

Population growth, compound interest, bacterial growth, viral spread.

Graph of y = (½)ˣ

When

0 < a < 1

, we get exponential decay.

x

-3-2-10123

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

8421

\frac{1}{2}

\frac{1}{4}

\frac{1}{8}

Key Features of y = (½)ˣ

y-intercept: Still (0, 1)

Horizontal asymptote: Still

y = 0

Decreasing function: as

x

increases,

y

decreases

This is exponential decay

Exponential Decay (0 < a < 1)

Important Relationship

(\frac{1}{a})^{x} = a^{- x}

So

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

is the same as

y = 2^{- x}

This means the graph of

y = (\frac{1}{a})^{x}

is a reflection of

y = a^{x}

in the y-axis.

Real-World Examples of Decay

Radioactive decay, cooling of objects, depreciation of value, drug concentration in bloodstream.

Key Properties Summary

Property

a > 1

(Growth)

0 < a < 1

(Decay)

y-intercept(0, 1)(0, 1)

Asymptote

y = 0

y = 0

Domain

x \in R

x \in R

Range

y > 0

y > 0

BehaviourIncreasingDecreasing

As

x \to + \infty

y \to + \infty

y \to 0

As

x \to - \infty

y \to 0

y \to + \infty

Remember

ALL exponential functions

y = a^{x}

pass through (0, 1) and have the x-axis as an asymptote. The value of

a

only determines whether it's growth or decay.

Worked Example01

Sketch the graph of $y = 2^{x}$ , clearly labelling the y-intercept and the equation of the asymptote.

1

Find the y-intercept by substituting

x = 0

:

y = 2^{0} = 1

, so y-intercept is (0, 1)

2

Identify the asymptote: As

x \to - \infty

,

y \to 0

Asymptote: $y = 0$ (the x-axis)

3

Since

a = 2 > 1

, this is exponential growth

4

Plot key points: (-1, 0.5), (0, 1), (1, 2), (2, 4)

5

Draw smooth curve through points, approaching y = 0 for negative x

Worked Example02

The curve $y = 5^{x}$ passes through the point $(k, 125)$ . Find the value of $k$ .

1

The point

(k, 125)

lies on the curve

y = 5^{x}

2

Substitute the point into the equation:

125 = 5^{k}

3

Express 125 as a power of 5:

125 = 5 \times 5 \times 5 = 5^{3}

4

So

5^{k} = 5^{3}

5

Comparing powers: $k = 3$

Answer

k = 3

Transformations of Exponential Graphs

We can apply transformations to the basic exponential function

y = a^{x}

:

TransformationEquationEffect

Vertical translation

y = a^{x} + k

Shifts up by

k

; asymptote becomes

y = k

Horizontal translation

y = a^{x - c}

Shifts right by

c

Vertical stretch

y = k a^{x}

Stretches by factor

k

; y-intercept becomes

(0, k)

Reflection in x-axis

y = - a^{x}

Asymptote still

y = 0

; range becomes

y < 0

Reflection in y-axis

y = a^{- x}

Converts growth to decay (or vice versa)

Important

When the graph is translated vertically, the asymptote also moves. A shift of

+ k

means the new asymptote is

y = k

.

Worked Example03

Sketch the graph of $y = 2^{x} + 3$ . State the equation of the asymptote and the coordinates of the y-intercept.

1

This is

y = 2^{x}

translated up by 3 units

2

New asymptote:

y = 3

(moved up from

y = 0

)

3

y-intercept: When

x = 0

:

y = 2^{0} + 3 = 1 + 3 = 4

y-intercept is (0, 4)

4

Key points: (-1, 3.5), (0, 4), (1, 5), (2, 7)

5

Range:

y > 3

(never touches or crosses the asymptote)

Worked Example04

Sketch the graph of $y = 2^{x - 1}$ . State the coordinates of the y-intercept and describe the transformation from $y = 2^{x}$ .

1

Transformation: Horizontal translation right by 1 unit

(Remember:

x - 1

means shift right)

2

Asymptote: Still

y = 0

(horizontal shifts don't affect the asymptote)

3

y-intercept: When

x = 0

:

y = 2^{0 - 1} = 2^{- 1} = \frac{1}{2}

y-intercept is (0, ½)

4

The point (0, 1) from

y = 2^{x}

moves to (1, 1)

5

Key points: (0, 0.5), (1, 1), (2, 2), (3, 4)

Practice

Question: Sketch the graph of

y = 4^{x}

. State the domain, range, and equation of the asymptote.

Graph: Exponential growth curve passing through (0, 1)

Domain:

x \in R

(all real numbers)

Range:

y > 0

Asymptote:

y = 0

Key points: (-1, ¼), (0, 1), (1, 4), (2, 16)

Since

4 > 1

, the function shows exponential growth (increasing).

Practice

Question: Sketch the graph of

y = (\frac{1}{3})^{x}

. State whether this represents exponential growth or decay, giving a reason.

Graph: Exponential decay curve passing through (0, 1)

Type: Exponential decay

Reason: Since

0 < \frac{1}{3} < 1

, the function is decreasing.

Alternatively:

(\frac{1}{3})^{x} = 3^{- x}

, which is a reflection of

y = 3^{x}

in the y-axis.

Key points: (-2, 9), (-1, 3), (0, 1), (1, ⅓), (2, ⅑)

Asymptote:

y = 0

Practice

Question: The graph of

y = 2^{x}

is transformed to give the graph of

y = 2^{x + 2} - 1

. Describe fully the transformation and state the equation of the new asymptote.

Transformations:

1. Horizontal translation left by 2 units (from

x + 2

)

2. Vertical translation down by 1 unit (from

- 1

)

New asymptote:

y = - 1

(shifted down from

y = 0

)

y-intercept: When

x = 0

:

y = 2^{0 + 2} - 1 = 4 - 1 = 3

, so (0, 3)

New range:

y > - 1

Summary

Key Concepts

ConceptDetails

Exponential Function

y = a^{x}

where

a > 0, a \neq = 1

Growth

a > 1

: increasing function

Decay

0 < a < 1

: decreasing function

Key Featuresy-intercept (0, 1), asymptote

y = 0

, range

y > 0

Relationship

(\frac{1}{a})^{x} = a^{- x}

(reflection in y-axis)

Exam Skills

Always label the y-intercept (0, 1) unless transformed

Always state the asymptote equation

Vertical translations change the asymptote; horizontal translations don't

Remember: larger base = steeper growth for

x > 0