1.4 Linear Functions and Slope
1.5 More on Slope
Objectives: Calculate slope and write linear
equations in different forms. Find slopes of
parallel and perpendicular lines.
Definition of Slope
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Example: Using the Definition of Slope
Find the slope of the line passing through the points
(4, –2) and (–1, 5)
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Possibilities for Line’s Slope
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Point-Slope Form of the Equation of a Line
The point-slope form of the equation of a nonvertical
line with slope m that passes through the point (x1, y1) is
y  y1  m( x  x1 ).
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Example: Writing an Equation in Point-Slope Form for a
Line
Write an equation in point-slope form for the line with
slope 6 that passes through the point (2, –5). Then solve
the equation for y.
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Slope-Intercept Form of the Equation of a Line
The slope-intercept form of the equation of a
nonvertical line with slope m and y-intercept b is
y  mx  b.
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Example: Graphing Using the Slope and y-Intercept
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Graph the linear function: f ( x)  x  1.
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Equation of a Horizontal Line
A horizontal line is given by an equation of the form
y = b,
where b is the y-intercept of the line.
The slope of a horizontal line is zero.
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Equation of a Vertical Line
A vertical line is given by an equation of the form
x = a,
where a is the x-intercept of the line.
The slope of a vertical line is undefined.
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Example: Graphing a Horizontal Line
Graph y = 3 in the rectangular coordinate system.
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General Form of the Equation of a Line
Every line has an equation that can be written in the
general form
Ax + By + C = 0
where A, B, and C are real numbers, and A and B are not
both zero.
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Example: Finding the Slope and the y-Intercept
Find the slope and the y-intercept of the line whose
equation is 3x  6 y  12  0.
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Using Intercepts to Graph Ax + By + C = 0
1. Find the x-intercept. Let y = 0 and solve for x. Plot
the point containing the x-intercept on the x-axis.
2. Find the y-intercept. Let x = 0 and solve for y. Plot
the point containing the y-intercept on the y-axis.
3. Use a straightedge to draw a line through the points
containing the intercepts. Draw arrowheads at the
ends of the line to show that the line continues
indefinitely in both directions.
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Example: Using Intercepts to Graph a Linear Equation
Graph using intercepts: 3x  2 y  6  0.
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A Summary of the Various Forms of Equations of Lines
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Example: Application
Use the data points (317, 57.04) and (354, 57.64) to
obtain a linear function that models average global
temperature, f(x), for an atmospheric carbon dioxide
concentration of x parts per million. Round m to three
decimal places and b to one decimal place.
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Slope and Parallel Lines
1. If two nonvertical lines are parallel, then they have
the same slope.
2. If two distinct nonvertical lines have the same slope,
then they are parallel.
3. Two distinct vertical lines, both with undefined
slopes, are parallel.
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Example: Writing Equations of a Line Parallel to a Given
Line
Write an equation of the line passing through (–2, 5)
and parallel to the line whose equation is y  3 x  1.
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Slope and Perpendicular Lines
1. If two nonvertical lines are perpendicular, then the
product of their slopes is –1.
2. If the product of the slopes of two lines is –1, then
the lines are perpendicular.
3. A horizontal line having zero slope is perpendicular
to a vertical line having undefined slope.
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Example: Writing Equations of a Line Perpendicular to a
Given Line
Find the slope of any line that is perpendicular to the
line whose equation is x  3 y  12  0
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Slope as Rate of Change
Slope is defined as the ratio of the change in y to a
corresponding change in x. It describes how fast y is
changing with respect to x. For a linear function, slope
may be interpreted as the rate of change of the
dependent variable per unit change in the independent
variable.
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Example: Slope as Rate of Change
In 1990, 9 million adult men in the United States lived
alone. In 2008, 14.7 million adult men in the United
States lived alone. Use this information to find the
slope of the linear function representing adult men
living alone in the United States. Express the slope
correct to two decimal places and describe what it
represents.
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The Average Rate of Change of a Function
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Example: Finding the Average Rate of Change
Find the average rate of change for f ( x)  x3
x1 = –2 to x2 = 0.
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Average Velocity of an Object
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Example: Finding Average Velocity
The distance, s(t), in feet, traveled by a ball rolling down a
ramp is given by the function
s (t )  4t 2
where t is the time, in seconds, after the ball is released. Find
the ball’s average velocity from t1 = 1 second to
t2 = 2 seconds.
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