Download Chapter 3: Linear Functions

Chapter 3: Linear Functions
3.1
Introduction to Linear Functions
I.
Definition

A linear function is a function whose general equation is y  mx  b where
m and b stand for constants, and m  0.

II.
Graph of Linear Function

III.
The number m is called the slope and determines the tilt of the line.
Effect of the value b

3.2
A typical graph of one of these functions looks like this:
Effect of the value m

IV.
Linear functions describe straight lines and have the general form
b is called the y-intercept, the point on the y-axis where the line crosses at
x = 0.
Properties of Linear Function Graphs
I.
Linear Function Graph Properties

Graphs are straight lines

Value of m determines tilt of graph
o positive m – graph slopes up as x increases
o negative m – graph slopes down as x increases
o m is zero – graph is horizontal (constant function)
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Chapter 3: Linear Functions
Linear Function Graph Properties (continued)

II.
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Value of b tells where the graph crosses the y-axis
Intercepts

y-intercept – the value of y when x = 0

x-intercept – the value of x when y = 0
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Chapter 3: Linear Functions
III.
Slope

rise
run

slope formula
m
y2  y1 y

x2  x1 x
How do you measure the slant of a line? By definition, it is the ratio of the vertical change to
the horizontal change (see figure below).
Forming the vertical change over the horizontal change (above) figure results in slope
formula (where m is the slope).
m= y - b
x-a

use this formula to calculate slopes of lines
WATCH OUT! Always use the same order in the numerator and denominator!
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Chapter 3: Linear Functions
Example
2
Graph the function f (x)   x 1 and determine its
3
slope.
Solution: Calculate two ordered pairs, plot the points,
graph the function, and determine its slope.
2
f (3)   (3) 1  2 1  1
3
2
f (9)    9 1  6 1  5
3
y2  y1
m
x2  x1
5  1 4
2

 
93
6
3
Copyright © 2009 Pearson Education, Inc.
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Page 4
Chapter 3: Linear Functions
IV.
Slope-Intercept Form

If y  mx  b , then m equals the slope of the graph and b equals the yintercept

Graphing Using Slope and Y-Intercept
Graph an equation using your knowledge of slope and y-intercept.
We can find the slope and y-intercept of the line just by looking at the
equation: m = 1/2 and y intercept = 2.
Just by looking at these values, we already know one point on the line! The yintercept gives us the point where the line intersects the y-axis, so we know
the coordinates of that point are (0, 2), since the x value of any point that lies
on the y axis is zero.
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Chapter 3: Linear Functions

Graphing Using Slope and Y-Intercept (continued)
To find the second point, we can use the slope of the line. The slope is ½, which
gives us the change in the y value over the change in the x value. The change in
the x value, the denominator, is 2, so we move to the right 2 units.
The change in the y value, the numerator, is positive one. We move up one
unit. This gives us the second point we need. Now we can draw the line
through the points.
Do you see that it's quicker and easier to use the y-intercept and the
slope?
Note: If you always use the y-axis as determining the
direction you will rise or fall according to the sign of the
slope, then you will ALWAYS run to the right on the x-axis!
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Chapter 3: Linear Functions
Example
2
x4
3
Solution: The equation is
in slope-intercept form,
y = mx + b.
Graph y  
The y-intercept is (0, 4).
Plot this point, then use
the slope to locate a
second point.
m
rise change in y 2


3
run change in x
 move 2 units down
 move 3 units right
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Copyright © 2009 Pearson Education, Inc.
V.
Horizontal and Vertical Lines

If y = constant, then the graph is a horizontal straight line with slope = 0
→The slope of any horizontal line is 0. In other words, as x increases or
decreases, y does not change.
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Chapter 3: Linear Functions

If x = constant, then the graph is a vertical straight line with undefined
slope
→The slope of any vertical line is undefined. x does not increase or
decrease; rather, y takes every possible value at a specific x value.
3.3
Other Forms of the Linear Function Equation

Point-Slope Form
→Coordinates of a point and the slope appear in the equation.
→A useful form is the point-slope form. We use this form when we need to find
the equation of a line passing through a point (x1, y1) with slope m:
y − y1 = m(x − x1)
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Chapter 3: Linear Functions

A
Standard Form
A linear equation in two variables is an equation that can be written in the form
Ax + By = C, where A and B are not both 0.
This form is called the standard form of a linear equation.
Determine whether the equation
y = 5x - 3 is linear or not.
If we subtract 5x from both sides, then we can write the given equation as -5x + y = -3.
Since we can write it in the standard form, Ax + By = C, then we have a linear equation.
If we were to graph this equation, we would end up with a graph of a straight line.
3.4
Equations of Linear Functions from Their Graphs
 Find the equation of a line given slope and a point on the line
o Since slope and a point are given, use the point-slope form
Watch Your Signs!!
Example
7
and y-intercept (0, 16). Find an
9
equation of the line.
Solution:
7
We use the slope-intercept equation and substitute 
9
for m and 16 for b:
y  mx  b
7
y 
x  16
9
or
A line has slope 
f
x  

7
x  16
9
Copyright © 2009 Pearson Education, Inc.
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Chapter 3: Linear Functions
Example
Find the equation of the line containing the points
(2, 3) and (1, 4).
Solution: First determine the slope
4  3
7
m

7
1 2
1
Using the point-slope
equation,
substitute 7 for m and
either of the
points for (x1, y1):
y  y1  m(x  x1 )
y  3  7(x  2)
y  3  7x  14
y  7x  11
or f x   7 x  11
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Parallel Lines
Vertical lines are parallel. Nonvertical lines are
parallel if and only if they have the same slope and
different y-intercepts.
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Chapter 3: Linear Functions
Perpendicular Lines
Two lines with slopes m1 and m2 are perpendicular if
and only if the product of their slopes is 1:
m1m2 = 1.
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Perpendicular Lines
Lines are also perpendicular if one is vertical (x = a)
and the other is horizontal (y = b).
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Chapter 3: Linear Functions
Example
Write equations of the lines (a) parallel and
(b) perpendicular to the graph of the line 4y – x = 20
and containing the point (2,  3).
Solution: Solve the equation for y:
4 y  x  20
y
1
x5
4
So the slope of this line is
Copyright © 2009 Pearson Education, Inc.
Example
1
.
4
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(continued)
(a) The line parallel to the given line will have the same
1
slope, . We use either the slope-intercept or point4
1
slope equation for the line. Substitute
for m and
4
use the point (2,  3) and solve the equation for y.
y  y1  m(x  x1 )
1
x  2 
4
1
1
y3 x
4
2
1
7
y x
4
2
y  3 
Copyright © 2009 Pearson Education, Inc.
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Chapter 3: Linear Functions
Example
(continued)
(b) The slope of the perpendicular line is the
1
negative reciprocal of , or – 4. Use the point4
slope equation, substitute – 4 for m and use the
point (2, –3) and solve the equation.
y  y1  m(x  x1 )
y  3  4 x  2 
y  3  4 x  8
y  4 x  5
Copyright © 2009 Pearson Education, Inc.
3.5
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Linear Functions as Mathematical Models

Given a situation in which two real-world variables are related by a straight-line graph:
1. Sketch the graph.
2. Find the equation of the line.
3. Use the equation to predict values of either variable.
4. Find the meaning of the slope and intercepts within the context of the problem.
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Chapter 3: Linear Functions
Mathematical Modeling
When a real-world problem can be described in a
mathematical language, we have a mathematical
model. The mathematical model gives results that
allow one to predict what will happen in that realworld situation. If the predictions are inaccurate or the
results of experimentation do not conform to the
model, the model must be changed or discarded.
Mathematical modeling can be an ongoing process.
Copyright © 2009 Pearson Education, Inc.
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Stat Plot and Linear Regression
Curve Fitting
In general, we try to find a function that fits, as well as
possible, observations (data), theoretical reasoning, and
common sense. We call this curve fitting, it is one
aspect of mathematical modeling.
In this chapter, we will explore linear relationships.
Let’s examine some data and related graphs, or scatter
plots and determine whether a linear function seems to
fit the data.
Copyright © 2009 Pearson Education, Inc.
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Chapter 3: Linear Functions
Linear Regression
Linear regression is a procedure that can be used
to model a set of data using a linear function.
We use the data on the number of U.S. households
with cable television.
We can fit a regression line of the form y = mx + b
to the data using the LINEAR REGRESSION
feature on a graphing calculator.
Copyright © 2009 Pearson Education, Inc.
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