Connect tables, graphs, and equations of exponential relations

Ontario Grade 11 Mathematics

Mathematical Models

MBF3C | Study-guide label: A2.5

An exponential relation links two quantities when one quantity changes by the same multiplier each time the other increases by one. You can show the relation with an equation, a table, or a graph. In this lesson, you will practise moving between those representations and checking that they agree. A multiplier can be greater than

1

, which gives growth, or between

0

and

1

, which gives decay.

What you will learn

Recognize a constant multiplicative change in a table.
Write an exponential equation from a starting value and a repeated multiplier.
Connect values in an exponential table to points and the shape of a graph.
Use an equation, table, and graph to describe the same situation.

1. Start with repeated change

A table lists paired values. The input is often called

x

, and the related output is often called

y

. Before studying exponential relations, recall the difference between adding the same amount and multiplying by the same amount.

If the outputs increase by a constant amount, the pattern is additive. For example, CAD 5, 8, 11 increases by

3

each time. In an exponential pattern, each output is found by multiplying the previous output by the same number. For example, CAD 5, 10, 20 doubles each time.

That repeated multiplier is called the common ratio. To find it, divide a value by the value immediately before it. Compare ratios only when the inputs move in equal steps, such as

1

unit at a time. A constant ratio signals an exponential relation in this setting.

\text{common ratio}=\frac{\text{next output}}{\text{previous output}}

Add the same amount for an additive pattern; multiply by the same amount for an exponential pattern.
A common ratio is found by dividing consecutive outputs.

2. Write the equation and read its parts

A useful equation for an exponential relation is

y=a(b)^x

. Here,

x

is the input and

y

is the output. The number

a

is the output when

x=0

, so it is the starting value. The number

b

is the multiplier for each one-unit increase in

x

. The exponent

x

tells how many times the multiplier is used.

For example, if a quantity starts at

12

and is multiplied by

1.5

for each step, its equation is

y=12(1.5)^x

. At

x=0

, the output is

12

because multiplying by the starting value's multiplier zero times leaves the starting value. At

x=1

, the output is

18

.

When

b>1

, the outputs increase as

x

increases; this is exponential growth. When

0<b<1

, the outputs decrease; this is exponential decay. In a percent-change situation, first convert the change to a multiplier. A 20% decrease leaves 80% of the amount, so the multiplier is

0.80

.

y=a(b)^x

The starting value is the output at $x=0$ .
The multiplier is applied once for each one-unit increase in the input.
A percent decrease is not itself the multiplier; use the percent that remains.

3. Connect a table to a graph

A graph displays each input-output pair as a point. For a table row with input

x=2

and output

y=27

, plot the point

(2,27)

. Repeat for the other rows. The horizontal axis usually shows the input, and the vertical axis shows the output.

For an exponential relation with a multiplier greater than

1

, the plotted points rise and typically become farther apart vertically as the input increases. With a multiplier between

0

and

1

, the points fall and get closer to the horizontal axis. A smooth curve can be drawn to show the overall pattern. The points come from the table; the curve helps show the trend between them.

Use all three representations to check your work. The equation should produce the table values. The graph should contain the points made from those values. If the points do not fit the curve's direction or pattern, check the equation, the table calculations, and the plotted coordinates.

(x,y)

Each table row gives one graph point.
The graph's direction should match whether the multiplier represents growth or decay.
The equation, table, and graph must describe the same input-output pairs.

4. Use the representations to explain a situation

A practical model often tells you the starting amount and how it changes repeatedly. The equation makes repeated change efficient to calculate. A table shows selected values clearly, and a graph makes the trend easy to see. The input must match the situation: if

x

counts hours, then moving from one table row to the next by one means one hour has passed.

Check that the units make sense. An output might be measured in people, grams, or dollars, while the input might be days or hours. The multiplier has no units because it compares two outputs measured in the same unit. A model's calculated values describe the relation; they should still be interpreted in the situation's context.

State what the input and output represent, including their units.
Use the equation to generate table values, then plot those pairs.
Check that the graph's trend and the context agree.

One relation in three forms

Representation	What it shows	How to check it
Equation	Starting value and repeated multiplier	Substitute an input and compare the output
Table	Several input-output pairs	Check for the same ratio between consecutive outputs
Graph	The plotted input-output pairs and overall trend	Check that plotted points match the table and curve direction

Worked example

Example 1: Find an equation from a table

A quantity is recorded at equal one-step intervals. The table shows outputs of CAD 12, 18, 27, 40.5, and

60.75

for inputs CAD 0, 1, 2, 3, and

4

. Find an exponential equation and describe how its graph connects to the table.

Find the repeated multiplier
Divide each output by the one before it. The quotients match, so the table has a constant multiplicative change rather than a constant additive change.
$\frac{18}{12}=\frac{27}{18}=\frac{40.5}{27}=\frac{60.75}{40.5}=1.5$
Identify the starting value
The output paired with input $0$ is the starting value. Here it is $12$ , so use $12$ for $a$ in the exponential equation.
$a=12$
Write the equation
The multiplier is $1.5$ , and it is used once for each increase of one in the input. Put the starting value and multiplier into the exponential form.
$y=12(1.5)^x$
Connect the table and graph
Plot each input-output pair as a point. Since the multiplier is greater than $1$ , the points rise from left to right. The equation reproduces the listed outputs, so the points and equation agree.
$(0,12),\ (1,18),\ (2,27),\ (3,40.5),\ (4,60.75)$

Answer: The equation is

y=12(1.5)^x

. Its graph passes through the table's points and rises from left to right.

Check: Substituting

x=3

gives

12(1.5)^3=12(3.375)=40.5

, which matches the table.

Worked example

Example 2: Connect a decay situation to all three forms

A sample has a mass of

160

grams. Each hour, it keeps 80% of its mass. Write an equation, make a table for the first three hours after the start, and describe the graph.

Convert the percent to a multiplier
Keeping 80% means multiplying by $0.80$ each hour. This is a decrease because the multiplier is less than $1$ .
$80\%=0.80$
Write the equation
At hour $0$ , the mass is $160$ grams, which is the starting value. Use $0.80$ as the repeated hourly multiplier.
$m=160(0.80)^t$
Calculate table values
Use the equation for each whole-number hour. Each new mass is the previous mass multiplied by $0.80$ .
$160(0.80)^1=128,\quad 160(0.80)^2=102.4,\quad 160(0.80)^3=81.92$
Describe the graph
Plot the time and mass pairs. The points descend as time increases because the mass is multiplied by a number below $1$ each hour.
$(0,160),\ (1,128),\ (2,102.4),\ (3,81.92)$

Answer: The equation is

m=160(0.80)^t

, where

t

is time in hours and

m

is mass in grams. The graph passes through the table's points and decreases from left to right.

Check: At

t=2

, the equation gives

160(0.64)=102.4

grams, matching the table. Each table output is 80% of the one before it.

Common mistakes and how to avoid them

Using the amount of a percent decrease as the multiplier.

Correction: For a 20% decrease, use the 80% that remains, or

0.80

, as the multiplier.

Treating the starting value as the output at input

1

.

Correction: The starting value is paired with input

0

. The output at input

1

is the starting value multiplied once.

Plotting the input and output on the wrong axes.

Correction: Use the input as the horizontal coordinate and the output as the vertical coordinate, in the order

(x,y)

.

Assuming any rising table is exponential.

Correction: Check the ratios between consecutive outputs when the inputs have equal steps. A constant difference alone does not establish an exponential pattern.

Lesson summary

An exponential relation has a repeated multiplier for equal steps in the input.
In $y=a(b)^x$ , $a$ is the output at input zero and $b$ is the repeated multiplier.
A table row $(x,y)$ becomes a point on the graph.
Check that the equation, table values, plotted points, and context agree.

Check your understanding

Question 1

A table has outputs CAD 9, 18, 36 at inputs CAD 0, 1, 2. Which equation matches it?

$y=9(2)^x$
$y=2(9)^x$
$y=9+2x$
correctIndex":0,"explanation":"The starting output is $9$ , and each output is twice the one before it. Thus the starting value is $9$ and the multiplier is $2$ ."

Show answer and explanation

y=9(2)^x

The starting output is

9

, and each output is twice the one before it. Thus the starting value is

9

and the multiplier is

2

.

Question 2

A value begins at

50

and decreases by 10% at each equal time step. What is its multiplier?

$0.10$
$0.90$
$1.10$
correctIndex":1,"explanation":"After a 10% $decrease,$ 90% remains. As a decimal multiplier, that is $0.90$ ."

Show answer and explanation

0.90

After a 10%

decrease,

90% remains. As a decimal multiplier, that is

0.90

.

Question 3

For the equation

y=6(1.4)^x

, which point must be on its graph?

$(0,1.4)$
$(1,6)$
$(0,6)$
correctIndex":2,"explanation":"At $x=0$ , the output is the starting value $6$ , so the graph includes $(0,6)$ ."

Show answer and explanation

(0,6)

At

x=0

, the output is the starting value

6

, so the graph includes

(0,6)

.

Key terms

Exponential relation: A relation in which the output is repeatedly multiplied by the same number for equal steps in the input.
Starting value: The output when the input is zero in an equation of the form $y=a(b)^x$ .
Multiplier: The number used to multiply one output to get the next output in equal input steps.
Common ratio: The quotient found by dividing a table output by the previous output.

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