Introduction
When linear functions are used to model real-world
relationships, the slope and y-intercept of the linear
function can be interpreted in context. Recall that data in
a scatter plot can be approximated using a linear fit, or
linear function that models real-world relationships. A
linear fit is the approximation of data using a linear
function.
1
4.3.1: Interpreting Slope and y-intercept
Introduction, continued
The slope of a linear function is the change in the
dependent variable divided by the change in the
change in y
independent variable, or
, sometimes written
change in x
Dy
as
.
Dx
2
4.3.1: Interpreting Slope and y-intercept
Introduction, continued
The slope between two points (x1, y1) and (x2, y2) is
y 2 - y1
, and the slope in the equation y = mx + b is m. The
x2 - x1
slope describes how much y changes when x changes
by 1. When analyzing the slope in the context of a realworld situation, remember to use the units of x and y in
the calculation of the slope.
3
4.3.1: Interpreting Slope and y-intercept
Introduction, continued
For example, if the x-axis of a graph represents hours
and the y-axis represents miles traveled, the slope of a
linear function graphed on these axes would be
change in miles
, or the miles traveled each hour.
change in hours
4
4.3.1: Interpreting Slope and y-intercept
Introduction, continued
The y-intercept of a function is the value of y at which
the graph of the function crosses the y-axis, or the value
of y when x equals 0. When analyzing the y-intercept in
a real-world context, this is the starting value of
whatever is represented by the y-axis. For example, if
the x-axis represents hours and the y-axis represents
miles traveled, the y-intercept would be the miles
traveled when the number of hours equals 0. The yintercept in the equation y = mx + b is b. In some cases,
the y-intercept doesn’t make sense in context, such as
when the quantity of x equals 0, and the y-intercept is
something other than 0 (see Example 2).
4.3.1: Interpreting Slope and y-intercept
5
Key Concepts
• The slope of a line with the equation y = mx + b is m.
• The slope of a line is
change in y
; the slope between
change in x
y - y1
two points (x1, y1) and (x2, y2) is 2
.
x2 - x1
• In context, the slope describes how much the
dependent variable changes each time the
independent variable changes by 1 unit.
6
4.3.1: Interpreting Slope and y-intercept
Key Concepts, continued
• The y-intercept of a line with the equation y = mx + b
is b.
• The y-intercept is the value of y at which a graph
crosses the y-axis.
• In context, the y-intercept is the initial value of the
quantity represented by the y-axis, or the quantity of y
when the quantity represented by the x-axis equals 0.
7
4.3.1: Interpreting Slope and y-intercept
Common Errors/Misconceptions
• incorrectly calculating the slope
• confusing the y- and x-intercepts, both in context and
when calculating using a graph or equation
8
4.3.1: Interpreting Slope and y-intercept
Guided Practice
Example 1
The graph on the next slide contains a linear model that
approximates the relationship between the size of a home
and how much it costs. The x-axis represents size in
square feet, and the y-axis represents cost in dollars.
Describe what the slope and the y-intercept of the linear
model mean in terms of housing prices.
9
4.3.1: Interpreting Slope and y-intercept
Cost in dollars ($)
Guided Practice: Example 1, continued
Size in square feet
4.3.1: Interpreting Slope and y-intercept
10
Guided Practice: Example 1, continued
1. Find the equation of the linear fit.
The general equation of a line in slope-intercept form
is y = mx + b, where m is the slope and b is the yintercept.
Find two points on the line using the graph.
The graph contains the points (300, 60,000) and
(600, 120,000).
11
4.3.1: Interpreting Slope and y-intercept
Guided Practice: Example 1, continued
The formula to find the slope between two points
y 2 - y1
(x1, y1) and (x2, y2) is
.
x2 - x1
Substitute (300, 60,000) and (600, 120,000) into the
formula to find the slope.
12
4.3.1: Interpreting Slope and y-intercept
Guided Practice: Example 1, continued
y 2 - y1
Slope formula
x2 - x1
=
120,000 - 60,000
600 - 300
60,000
300
= 200
Substitute (300, 60,000) and
(600, 120,000) for (x1, y1) and
(x2, y2).
Simplify as needed.
The slope between the two points (300, 60,000) and
(600, 120,000) is 200.
13
4.3.1: Interpreting Slope and y-intercept
Guided Practice: Example 1, continued
Find the y-intercept. Use the equation for slopeintercept form, y = mx + b, where b is the y-intercept.
Replace x and y with values from a single point on
the line. Let’s use (300, 60,000).
Replace m with the slope, 200. Solve for b.
14
4.3.1: Interpreting Slope and y-intercept
Guided Practice: Example 1, continued
y = mx + b
Equation for slope-intercept form
60,000 = 200(300) + b Substitute values for x, y, and m.
60,000 = 60,000 + b
Multiply.
0=b
Subtract 60,000 from both sides.
The y-intercept of the linear model is 0.
The equation of the line is y = 200x.
15
4.3.1: Interpreting Slope and y-intercept
Guided Practice: Example 1, continued
2. Determine the units of the slope.
Divide the units on the y-axis by the units on the
x-axis:
dollars
square feet
.
The units of the slope are dollars per square foot.
16
4.3.1: Interpreting Slope and y-intercept
Guided Practice: Example 1, continued
3. Describe what the slope means in
context.
The slope is the change in cost of the home for
each square foot of the home. The slope describes
how price is affected by the size of the home
purchased. A positive slope means the quantity
represented by the y-axis increases when the
quantity represented by the x-axis also increases.
The cost of the home increases by $200 for each
square foot.
17
4.3.1: Interpreting Slope and y-intercept
Guided Practice: Example 1, continued
4. Determine the units of the y-intercept.
The units of the y-intercept are the units of the
y-axis: dollars.
18
4.3.1: Interpreting Slope and y-intercept
Guided Practice: Example 1, continued
5. Describe what the y-intercept means in
context.
The y-intercept is the value of the equation when
x = 0, or when the size of the home is 0 square feet.
For a home with no area, or for no home, the cost is
$0.
✔
19
4.3.1: Interpreting Slope and y-intercept
Guided Practice: Example 1, continued
20
4.3.1: Interpreting Slope and y-intercept
Guided Practice
Example 2
A teller at a bank records the amount of time a customer
waits in line and the number of people in line ahead of
that customer when he or she entered the line. Describe
the relationship between waiting time and the people
ahead of a customer when the customer enters a line.
Use the table on the following slide.
21
4.3.1: Interpreting Slope and y-intercept
Guided Practice: Example 2, continued
People ahead of customer
Minutes waiting
1
10
2
21
3
32
5
35
8
42
9
45
10
61
22
4.3.1: Interpreting Slope and y-intercept
Guided Practice: Example 2, continued
1. Create a scatter plot of the data.
Let the x-axis represent the number of people ahead
of the customer and the y-axis represent the minutes
spent waiting.
23
4.3.1: Interpreting Slope and y-intercept
Minutes spent waiting
Guided Practice: Example 2, continued
Number of people ahead
4.3.1: Interpreting Slope and y-intercept
24
Guided Practice: Example 2, continued
2. Find the equation of a linear model to
represent the data.
Use two points to estimate a linear model. A line
through the two points should have approximately
the same number of data values both above and
below the line. A line through the first and last data
points, (1, 10) and (10, 61), appears to be a good
approximation of the data. Find the slope.
25
4.3.1: Interpreting Slope and y-intercept
Guided Practice: Example 2, continued
The slope between two points (x1, y1) and (x2, y2) is
y 2 - y1
. Substitute the points into the formula to find the
x2 - x1
slope.
26
4.3.1: Interpreting Slope and y-intercept
Guided Practice: Example 2, continued
y 2 - y1
x2 - x1
Slope formula
61- 10
Substitute (1, 10) and (10, 61) for
(x1, y1) and (x2, y2).
10 - 1
51
9
» 5.67
Simplify as needed.
The slope between the two points (1, 10) and
(10, 61) is approximately 5.67.
27
4.3.1: Interpreting Slope and y-intercept
Guided Practice: Example 2, continued
Find the y-intercept. Use the equation for slopeintercept form, y = mx + b, where b is the y-intercept.
Replace x and y with values from a single point on
the line. Let’s use (1, 10).
Replace m with the slope, 5.67. Solve for b.
28
4.3.1: Interpreting Slope and y-intercept
Guided Practice: Example 2, continued
y = mx + b
Equation for slope-intercept form
10 = 1(5.67) + b
Substitute values for x, y, and m.
10 = 5.67 + b
Simplify.
4.33 = b
Subtract 5.67 from both sides.
The y-intercept of the linear model is 4.33.
The equation of the line is y = 5.67x + 4.33.
29
4.3.1: Interpreting Slope and y-intercept
Guided Practice: Example 2, continued
3. Determine the units of the slope.
Divide the units on the y-axis by the units on the
minutes spent waiting
minutes
=
x-axis:
.
number of people ahead person
The units of the slope are minutes per person.
30
4.3.1: Interpreting Slope and y-intercept
Guided Practice: Example 2, continued
4. Describe what the slope means in
context.
The slope describes how the waiting time increases
for each person in line ahead of the customer. A
customer waits approximately 5.67 minutes for each
person who is in line ahead of the customer.
31
4.3.1: Interpreting Slope and y-intercept
Guided Practice: Example 2, continued
5. Determine the units of the y-intercept.
The units of the y-intercept are the units of the
y-axis: minutes.
32
4.3.1: Interpreting Slope and y-intercept
Guided Practice: Example 2, continued
6. Describe what the y-intercept means in
context.
The y-intercept is the value of the equation when
x = 0, or when the number of people ahead of the
customer is 0. The y-intercept is 4.33. In this
context, the y-intercept isn’t relevant, because if no
one was in line ahead of a customer, the wait time
would be 0 minutes. Creating a linear model that
matched the data resulted in a y-intercept that
wasn’t 0, but this value isn’t related to the
context of the situation.
✔
33
4.3.1: Interpreting Slope and y-intercept
Guided Practice: Example 2, continued
34
4.3.1: Interpreting Slope and y-intercept