Introduction
Tables and graphs can be represented by equations.
Data represented in a table can either be analyzed as a
pattern, like the data presented in the previous lesson, or
it can be graphed first. The shape of a graph helps
identify which type of equation can be used to represent
the data. Data that is represented using a linear equation
has a constant slope and the graph is a straight line.
1
3.6.2: Constructing Functions from Graphs and Tables
Introduction, continued
The general form of a linear equation is y = mx + b,
where m is the slope and b is the y-intercept. The yintercept is the point at which the graph intersects the yaxis. To find the equation of a line, first find the slope,
then replace x and y in the general equation with x and y
from an ordered pair on the graph. Then, solve the
equation for b.
2
3.6.2: Constructing Functions from Graphs and Tables
Introduction, continued
The slope of a line is the change in the independent
variable divided by the change in the dependent variable,
and describes the rate of change of the variables. Using
the points (x1, y1) and (x2, y2), the slope can be calculated
y 2 - y1)
(
by finding
. The slope is similar to the common
( x2 - x1)
difference between terms, except the slope can be found
between points that have x-values that are more than
one unit apart.
3
3.6.2: Constructing Functions from Graphs and Tables
Introduction, continued
The slope allows us to compare the change between any
two points on a graph or in a table. An exponential
equation has a slope that is constantly changing, either
increasing or decreasing. Graphically, exponential
equations are a curve. The general form of an
exponential equation is y = ab x, where a and b are real
numbers, and b is the common ratio. The values of a and
b change the shape of the graph of an exponential
function. On the following slides, you’ll find four examples
of functions with different a and b values.
4
3.6.2: Constructing Functions from Graphs and Tables
Introduction, continued
a > 0 and b > 1
f (x) = 2 x
5
3.6.2: Constructing Functions from Graphs and Tables
Introduction, continued
a > 0 and 0 < b < 1
f (x) = (0.5) x
6
3.6.2: Constructing Functions from Graphs and Tables
Introduction, continued
a < 0 and b > 1
f (x) = (–1) • 2x
7
3.6.2: Constructing Functions from Graphs and Tables
Introduction, continued
a < 0 and 0 < b < 1
æ 1ö
f (x) = -1ç ÷
è 2ø
x
8
3.6.2: Constructing Functions from Graphs and Tables
Introduction, continued
Data that follows an exponential pattern has a common
ratio between the dependent quantities. When looking for
the common ratio, be sure to verify that the x-values, or
independent quantities, are each one unit apart. The
common ratio, b, can be used to write an equation to
represent the exponential pattern. Find the value of the
equation at x = 0, f (0). The general equation of the
function is f (x) = f (0) • bx.
9
3.6.2: Constructing Functions from Graphs and Tables
Key Concepts
• The graph of a linear equation is a straight line.
• Linear equations have a constant slope, or rate of
change.
• Linear equations can be written as functions.
• The general form of a linear function is f (x) = mx + b,
where m is the slope and b is the y-intercept.
• The slope of a linear function can be calculated using
any two points, (x1, y1) and (x2, y2): the formula is
(y
(x
2
2
- y1)
- x1 ) .
3.6.2: Constructing Functions from Graphs and Tables
10
Key Concepts, continued
• The y-intercept is the point at which the graph of the
equation crosses the y-axis.
• Exponential equations have a slope that is constantly
changing.
• Exponential equations can be written as functions.
• The general form of an exponential function is
f(x) = ab x, where a and b are real numbers.
11
3.6.2: Constructing Functions from Graphs and Tables
Key Concepts, continued
• The graph of an exponential equation is a curve.
• The common ratio, b, between independent quantities
in an exponential pattern, and the value of the
equation at x = 0, f(0), can be used to write the
general equation of the function: f(x) = f (0) • b x.
12
3.6.2: Constructing Functions from Graphs and Tables
Common Errors/Misconceptions
• looking for a common difference between dependent
quantities when the independent values are not one
unit apart
• incorrectly calculating the slope
• trying to match an exponential graph to a linear
equation
• trying to match a linear graph to an exponential
equation
13
3.6.2: Constructing Functions from Graphs and Tables
Guided Practice
Example 2
Determine the
equation that
represents the
relationship between
x and y in the graph
to the right.
14
3.6.2: Constructing Functions from Graphs and Tables
Guided Practice: Example 2, continued
1. Determine which type of equation, linear or
exponential, will fit the graph.
The graph of a linear equation is a straight line, and a
graph of an exponential equation is a curve. A linear
equation can be used to represent the graph.
15
3.6.2: Constructing Functions from Graphs and Tables
Guided Practice: Example 2, continued
2. Identify at least three points from the
graph.
From the graph, three points are: (0, –4), (1, –1),
and (2, 2).
16
3.6.2: Constructing Functions from Graphs and Tables
Guided Practice: Example 2, continued
3. Find the slope of the line, using any two
of the points.
The slope of the line is
(y
(x
2
2
- y1)
- x1 )
for any two points
(x1, y1) and (x2, y2). Using the points (0, –4) and
-1- ( -4) )
(
(1, –1), the slope is
= 3.
(1- 0)
17
3.6.2: Constructing Functions from Graphs and Tables
Guided Practice: Example 2, continued
4. Find the y-intercept of the line.
The y-intercept can either be found by solving the
equation f(x) = mx + b for b, or by finding the value
of y when x = 0. On the graph, we can see the
point (0, –4). The y-intercept is –4.
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3.6.2: Constructing Functions from Graphs and Tables
Guided Practice: Example 2, continued
5. Use the slope and y-intercept to find an
equation of the line.
The general form of the linear function is f (x) = mx +
b, where m is the slope and b is the y-intercept. The
equation to represent the line is f (x) = 3x – 4. Check
the equation by evaluating it at the values of x from
previously identified points.
x = 0, f (0) = 3(0) – 4 = –4
x = 1, f (1) = 3(1) – 4 = –1
x = 2, f (2) = 3(2) – 4 = 2
✔
The relationship can be represented using the
equation f (x) = 3x – 4.
3.6.2: Constructing Functions from Graphs and Tables
19
Guided Practice: Example 2, continued
20
3.6.2: Constructing Functions from Graphs and Tables
Guided Practice
Example 3
A clothing store discounts items on a regular schedule.
Each week, the price of
Week
Price in dollars ($)
an item is reduced. The
0
100.00
prices for one item are in
1
60.00
the table to the right.
2
36.00
Week 0 is the starting
3
21.60
price of the item.
Determine a linear or exponential equation that
represents the relationship between the week and the
price of the item.
3.6.2: Constructing Functions from Graphs and Tables
21
Guided Practice: Example 3, continued
1. Create a graph of the data.
Let the x-axis
represent the
week, and the
y-axis represent
the price in dollars.
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3.6.2: Constructing Functions from Graphs and Tables
Guided Practice: Example 3, continued
2. Determine if a linear or exponential
equation could represent the data.
The x-values of the points vary by 1. Look at the
vertical distance between each pair of points. It
appears to be decreasing, and is not remaining
constant. An exponential equation could represent
the data.
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3.6.2: Constructing Functions from Graphs and Tables
Guided Practice: Example 3, continued
3. Find the common ratio between the
terms.
The four points are at the x-values 0, 1, 2, and 3, so
the x-values are each one unit apart. Look at the
pattern of the y-values: 100, 60, 36, and 21.60.
Divide each y-value by the previous y-value to
identify the common ratio.
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3.6.2: Constructing Functions from Graphs and Tables
Guided Practice: Example 3, continued
y
Ratio
100
60
36
21.6
60
100
36
= 0.60
= 0.60
60
21.6
36
= 0.60
The common ratio is 0.60.
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3.6.2: Constructing Functions from Graphs and Tables
Guided Practice: Example 3, continued
4. Use the value of the equation at x = 0 and
the common ratio to write an equation to
represent the relationship.
At week 0, the price is $100. The common ratio is
0.60. An equation to represent the relationship is
f(x) = 100 • (0.60)x. Evaluate the equation at the
given values of x to check the equation.
x = 0, f (0) = 100 • (0.60)0 = 100
x = 1, f (1) = 100 • (0.60)1 = 60
x = 2, f (2) = 100 • (0.60)2 = 36
x = 3, f (3) = 100 • (0.60)3 = 21.6
26
3.6.2: Constructing Functions from Graphs and Tables
The price of the clothing item, y, at any week, x, can
be represented by the equation f (x) = 100 • (0.60) x.
✔
27
3.6.2: Constructing Functions from Graphs and Tables
Guided Practice: Example 3, continued
28
3.6.2: Constructing Functions from Graphs and Tables