Graph simple exponential relations

Ontario Grade 11 Mathematics

Mathematical Models

Use a table, recognize a repeated multiplier, and sketch the curve

A quantity that changes by the same amount each step grows or shrinks by addition. A quantity that changes by the same factor each step grows or shrinks by multiplication. This second pattern can be represented by an exponential relation. In this lesson, you will use a rule and a table to graph simple exponential relations and connect the graph’s shape to the repeated multiplication in the rule.

What you will learn

Recognize a simple exponential relation from its rule or a table.
Make a table of values and plot its ordered pairs.
Sketch a smooth graph and describe whether it shows growth or decay.

1. Review: coordinates and tables

A graph shows how two quantities are related. The horizontal axis shows the input, often called

x

. The vertical axis shows the output, often called

y

. An ordered pair, such as

(2,12)

, means that the input is

2

and the output is

12

.

A table helps organize inputs and outputs before you draw a graph. To graph a table, plot each ordered pair where its horizontal and vertical coordinates meet. A smooth curve can then show the pattern between the plotted points.

Each table row gives one ordered pair.
Keep the horizontal and vertical scales even, and label both axes.

2. What makes a relation exponential?

An exponential relation has a variable in the exponent. A common simple form is

y=a(b)^x

. Here,

a

is the output when

x=0

, and

b

is the factor that multiplies the output each time

x

increases by

1

. The number

b

is called the multiplier or base.

For example, if the multiplier is

2

, the outputs double at each step. If the multiplier is

0.5

, the outputs are halved at each step. A multiplier greater than

1

shows growth as

x

increases. A positive multiplier less than

1

shows decay, meaning the outputs get smaller as

x

increases.

The value at

x=0

is useful when making a table: any non-zero number raised to the power of zero is

1

, so the output at that input is

a

. For positive whole-number inputs, continue by multiplying the previous output by

b

. This repeated-factor pattern distinguishes exponential change from a constant increase or decrease.

y=a(b)^x

In $y=a(b)^x$ , $a$ gives the output at $x=0$ .
The multiplier $b$ tells how the output changes for each increase of $1$ in $x$ .
A multiplier above $1$ gives growth; a multiplier between $0$ and $1$ gives decay.

3. From a table to a graph

Choose a few input values that make the rule easy to calculate. Find the output for each input and record the ordered pairs. Plot the pairs using a scale that fits all of them. Then connect them with a smooth curve that follows the plotted pattern; do not join them with separate straight-line segments.

For growth, the curve rises as it moves to the right and becomes steeper for the examples in this lesson. For decay, the curve falls as it moves to the right and gets closer to the horizontal axis. The graph helps you see the overall pattern, while the table gives exact values at its listed inputs.

If a relation comes from a situation, the input values may be limited by what the input represents. For example, elapsed time cannot be negative in a model that begins at time zero. Label the axes with the quantities and units from the situation, and graph only inputs that make sense there.

Calculate and plot several ordered pairs before sketching.
Use a smooth curve that follows the points.
Read the axes and any practical limits in the context.

4. Use the graph to describe the pattern

A graph is more than a set of points. Its direction shows whether the outputs increase or decrease as the input increases. Its steepness shows how quickly the outputs are changing over the part of the graph you can see.

Check a graph against its table before you finish. Each plotted point should match its ordered pair, and the curve should follow the pattern of the outputs. In a growth table, successive outputs are multiplied by the same factor greater than

1

. In a decay table, they are multiplied by the same positive factor less than

1

.

For a practical graph, state what the coordinates mean. A point such as

(3,40)

might mean that after

3

time units, the quantity is

40

units. The labels and context are needed to interpret that point correctly.

Use the table to verify plotted points.
Describe growth or decay by comparing outputs as the input increases.
Interpret points using the axis labels and units.

Values for the growth example

$x$	$y$	Ordered pair
0	3	$(0,3)$
1	6	$(1,6)$
2	12	$(2,12)$
3	24	$(3,24)$
4	48	$(4,48)$

Worked example

Example 1: Graph a growth relation

Make a table and sketch the graph of

y=3(2)^x

for

x=0,1,2,3,4

. Describe the pattern.

Find the starting value
At $x=0$ , the power is zero, so the exponential factor is $1$ . The output is therefore the starting value, $3$ .
$y=3(2)^0=3$
Continue the table
Each time $x$ increases by $1$ , multiply the previous output by $2$ . This follows from the rule’s repeated factor.
$3,\ 6,\ 12,\ 24,\ 48$
Plot and sketch
Plot the ordered pairs from the table. Draw a smooth curve through them. Since the outputs double at each step, the curve rises more quickly as it moves right.
$(0,3),\ (1,6),\ (2,12),\ (3,24),\ (4,48)$

Answer: The outputs are

3

,

6

,

12

,

24

, and

48

. The graph shows growth because the output doubles whenever the input increases by

1

.

Check: Every consecutive pair of outputs has a ratio of

2

, so the table matches the multiplier in the rule.

Worked example

Example 2: Graph a decay relation in context

A sample begins with

80

grams of a substance. The amount is multiplied by

0.5

each hour. Make a table for the first four whole-number hour values, including hour zero, and describe the graph.

Write the relation
The initial amount is $80$ grams, and the repeated multiplier is $0.5$ . Let $t$ represent time in hours and $A$ represent the amount in grams.
$A=80(0.5)^t$
Calculate the amounts
At hour zero, the amount is $80$ . For each following hour, multiply the previous amount by $0.5$ , which halves it.
$80,\ 40,\ 20,\ 10$
Plot and describe
Plot the points with time on the horizontal axis and amount on the vertical axis. The smooth curve falls as time increases because each new amount is half the previous one.
$(0,80),\ (1,40),\ (2,20),\ (3,10)$

Answer: The amounts are

80

,

40

,

20

, and

10

grams at hours

0

,

1

,

2

, and

3

. The graph shows decay.

Check: The amount at each listed hour is half the amount at the previous hour, matching the multiplier

0.5

.

Common mistakes and how to avoid them

Adding the multiplier to find the next output.

Correction: The multiplier tells you to multiply the previous output. For a multiplier of

2

, double the output each step.

Plotting the input on the vertical axis.

Correction: Put the input on the horizontal axis and the output on the vertical axis.

Joining plotted points with separate straight segments.

Correction: Sketch a smooth curve that follows the exponential pattern through the points.

Calling a relation growth because its outputs are positive.

Correction: Check how the outputs change as the input increases. A multiplier greater than

1

gives growth; a positive multiplier below

1

gives decay.

Lesson summary

A simple exponential relation can be written as $y=a(b)^x$ .
The value $a$ is the output at input zero, and $b$ is the repeated multiplier.
Make a table, plot ordered pairs, and sketch a smooth curve.
A multiplier above $1$ shows growth; a positive multiplier below $1$ shows decay.

Check your understanding

Question 1

For

y=5(0.5)^x

, what are the outputs when

x=0,1,2

?

5,\ 5.5,\ 6
5,\ 2.5,\ 1.25
0,\ 5,\ 10
5,\ 10,\ 20

Show answer and explanation

5,\ 2.5,\ 1.25

The output at zero is

5

. Multiply by

0.5

for each increase of

1

in

x

, giving

2.5

and then

1.25

.

Question 2

A table has outputs

4

,

12

, and

36

at inputs

0

,

1

, and

2

. What multiplier does the pattern use?

$3$
$8$
$4$
$0.5$

Show answer and explanation

3

Each output is three times the previous one:

12

is

4

multiplied by

3

, and

36

is

12

multiplied by

3

.

Key terms

Exponential relation: A relation in which the variable appears as an exponent, commonly written in the form $y=a(b)^x$ .
Multiplier: The factor by which an output is multiplied when the input increases by $1$ .
Ordered pair: A pair of coordinates written as $(x,y)$ that identifies one point on a graph.
Growth: A pattern in which the outputs increase as the input increases.
Decay: A pattern in which the outputs decrease as the input increases.

Continue through MBF3C

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About this lesson and its review

Published by DoAssignment. This reviewed lesson follows Ontario Grade 11 Mathematics (MBF3C), expectation A2.4. It is a study resource, not an official curriculum publication.

Before publication, content is checked for structure, mathematical or chemical notation, calculations, course boundaries, and readability. Errors can still occur, so corrections are welcomed.

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