Transcript 2-3 Graphing Linear Functions

2-3
2-3 Graphing
GraphingLinear
LinearFunctions
Functions
Warm Up
Lesson Presentation
Lesson Quiz
Holt McDougal
Algebra 2 Algebra 2
2-3 Graphing Linear Functions
Warm Up
Solve each equation for y.
1. 7x + 2y = 6
2.
y = –2x – 8
3. If 3x = 4y + 12, find y when x = 0. y = –3
4. If a line passes through (–5, 0) and (0, 2), then it
passes through all but which quadrant.
IV
Holt McDougal Algebra 2
2-3 Graphing Linear Functions
Objectives
Determine whether a function is linear.
Graph a linear function given two
points, a table, an equation, or a point
and a slope.
Holt McDougal Algebra 2
2-3 Graphing Linear Functions
Vocabulary
linear function
slope
y-intercept
x-intercept
slope-intercept form
Holt McDougal Algebra 2
2-3 Graphing Linear Functions
Meteorologists begin
tracking a hurricane's
distance from land
when it is 350 miles
off the coast of Florida
and moving steadily
inland.
The meteorologists are interested in the rate
at which the hurricane is approaching land.
Holt McDougal Algebra 2
2-3 Graphing Linear Functions
+1
Time (h)
Distance from Land (mi)
0
1
+1
+1
2
3
4
350 325 300 275 250
–25
This rate can be expressed as
+1
–25
–25
–25
.
Notice that the rate of change is constant. The hurricane
moves 25 miles closer each hour.
Holt McDougal Algebra 2
2-3 Graphing Linear Functions
Functions with a constant rate of change are
called linear functions. A linear function can be
written in the form f(x) = mx + b, where x is the
independent variable and m and b are constants.
The graph of a linear function is a straight line
made up of all points that satisfy y = f(x).
Holt McDougal Algebra 2
2-3 Graphing Linear Functions
Example 1A: Recognizing Linear Functions
Determine whether the data set could
represent a linear function.
+2
+2 +2
x
–2
0
2
4
f(x)
2
1
0
–1
–1
–1
The rate of change,
So the data set is linear.
Holt McDougal Algebra 2
–1
, is constant
.
2-3 Graphing Linear Functions
Example 1B: Recognizing Linear Functions
Determine whether the data set could
represent a linear function.
+1
+1 +1
x
2
3
4
5
f(x)
2
4
8
16
+2
The rate of change,
+4
+8
, is not constant.
2 ≠ 4 ≠ 8. So the data set is not linear.
Holt McDougal Algebra 2
2-3 Graphing Linear Functions
Check It Out! Example 1A
Determine whether the data set could
represent a linear function.
+7
+7
+7
x
f(x)
4
11
18
–6 –15 –24 –33
–9
–9
The rate of change,
So the data set is linear.
Holt McDougal Algebra 2
25
–9
, is constant
.
2-3 Graphing Linear Functions
Check It Out! Example 1B
Determine whether the data set could
represent a linear function.
–2
–2
–2
x
10
8
6
4
f(x)
7
5
1
–7
–4
The rate of change,
–4
–8
, is not constant.
. So the data set is not linear.
Holt McDougal Algebra 2
2-3 Graphing Linear Functions
The constant rate of change for a linear
function is its slope. The slope of a linear
function is the ratio
, or
.
The slope of a line is the same between
any two points on the line. You can graph
lines by using the slope and a point.
Holt McDougal Algebra 2
2-3 Graphing Linear Functions
Example 2A: Graphing Lines Using Slope and a Point
Graph the line with slope
through (–1, –3).
Plot the point (–1, –3).
The slope indicates a rise of 5
and a run of 2. Move up 5 and
right 2 to find another point.
Then draw a line through the
points.
Holt McDougal Algebra 2
that passes
2-3 Graphing Linear Functions
Example 2B: Graphing Lines Using Slope and a Point
Graph the line with slope
through (0, 2).
Plot the point (0, 2).
The negative slope can be
viewed as
You can move down 3 units
and right 4 units, or move up
3 units and left 4 units.
Holt McDougal Algebra 2
that passes
2-3 Graphing Linear Functions
Check It Out! Example 2
Graph the line with slope
through (3, 1).
Plot the point (3, 1).
The slope indicates a rise of 4
and a run of 3. Move up 4 and
right 3 to find another point.
Then draw a line through the
points.
Holt McDougal Algebra 2
that passes
2-3 Graphing Linear Functions
Recall from geometry that
two points determine a line.
Often the easiest points to
find are the points where a
line crosses the axes.
The y-intercept is the
y-coordinate of a point
where the line crosses the
x-axis.
The x-intercept is the
x-coordinate of a point where
the line crosses the y-axis.
Holt McDougal Algebra 2
2-3 Graphing Linear Functions
Example 3: Graphing Lines Using the Intercepts
Find the intercepts of 4x – 2y = 16, and graph
the line.
Find the x-intercept: 4x – 2y = 16
4x – 2(0) = 16 Substitute 0 for y.
4x = 16
The x-intercept is 4.
x=4
x-intercept
Find the y-intercept: 4x – 2y = 16
4(0) – 2y = 16 Substitute 0 for x.
–2y = 16
y = –8
Holt McDougal Algebra 2
The y-intercept is –8.
y-intercept
2-3 Graphing Linear Functions
Check It Out! Example 3
Find the intercepts of 6x – 2y = –24, and
graph the line.
Find the x-intercept: 6x – 2y = –24
6x – 2(0) = –24 Substitute 0 for y.
6x = –24
x = –4 The x-intercept is –4.
Find the y-intercept: 6x – 2y = –24
6(0) – 2y = –24 Substitute 0 for x.
y-intercept
x-intercept
–2y = –24
y = 12 The y-intercept is 12.
Holt McDougal Algebra 2
2-3 Graphing Linear Functions
Linear functions can also be expressed as linear
equations of the form y = mx + b. When a linear
function is written in the form y = mx + b, the
function is said to be in slope-intercept form
because m is the slope of the graph and b is the
y-intercept. Notice that slope-intercept form is the
equation solved for y.
Holt McDougal Algebra 2
2-3 Graphing Linear Functions
Example 4A: Graphing Functions in Slope-Intercept
Form
Write the function –4x + y = –1 in slope-intercept
form. Then graph the function.
Solve for y first.
–4x + y = –1
+4x
+4x Add 4x to both sides.
y = 4x – 1
The line has y-intercept –1 and
slope 4, which is . Plot the point
(0, –1). Then move up 4 and
right 1 to find other points.
Holt McDougal Algebra 2
2-3 Graphing Linear Functions
Example 4A Continued
You can also use a graphing
calculator to graph. Choose
the standard square
window to make your graph
look like it would on a
regular grid. Press ZOOM,
choose 6:ZStandard,
press ZOOM again, and
then choose 5:ZSquare.
Holt McDougal Algebra 2
2-3 Graphing Linear Functions
Example 4B: Graphing Functions in Slope-Intercept
Form
Write the function
in slope-intercept
form. Then graph the function.
Solve for y first.
Multiply both sides by
Distribute.
The line has y-intercept 8 and slope
.
Plot the point (0, 8). Then move down 4
and right 3 to find other points.
Holt McDougal Algebra 2
2-3 Graphing Linear Functions
Check It Out! Example 4A
Write the function 2x – y = 9 in slope-intercept
form. Then graph the function.
Solve for y first.
2x – y = 9
–2x
–2x Add –2x to both sides.
–y = –2x + 9
y = 2x – 9 Multiply both
sides by –1.
The line has y-intercept –9 and
slope 2, which is . Plot the point
(0, –9). Then move up 2 and right 1
to find other points.
Holt McDougal Algebra 2
2-3 Graphing Linear Functions
Check It Out! Example 4A Continued
You can also use a graphing
calculator to graph. Choose
the standard square
window to make your graph
look like it would on a
regular grid. Press ZOOM,
choose 6:ZStandard,
press ZOOM again, and
then choose 5:ZSquare.
Holt McDougal Algebra 2
2-3 Graphing Linear Functions
Check It Out! Example 4B
Write the function 5x = 15y + 30 in slopeintercept form. Then graph the function.
Solve for y first.
5x = 15y + 30
–30
–30 Subtract 30 from both sides.
5x – 30 = 15y
Divide both sides by 15.
The line has y-intercept –2 and slope .
Plot the point (0, –2). Then move up 1
and right 3 to find other points.
Holt McDougal Algebra 2
2-3 Graphing Linear Functions
An equation with only one variable can
be represented by either a vertical or a
horizontal line.
Holt McDougal Algebra 2
2-3 Graphing Linear Functions
Vertical and Horizontal Lines
Vertical Lines
Horizontal Lines
The line x = a is a vertical
line at a.
The line y = b is a vertical
line at b.
Holt McDougal Algebra 2
2-3 Graphing Linear Functions
The slope of a vertical line is undefined.
The slope of a horizontal line is zero.
Holt McDougal Algebra 2
2-3 Graphing Linear Functions
Example 5: Graphing Vertical and Horizontal Lines
Determine if each line is vertical or horizontal.
A. x = 2
This is a vertical line
located at the x-value 2.
(Note that it is not a
function.)
B. y = –4
This is a horizontal line
located at the y-value –4.
Holt McDougal Algebra 2
x=2
y = –4
2-3 Graphing Linear Functions
Check It Out! Example 5
Determine if each line is vertical or horizontal.
A. y = –5
This is a horizontal line
located at the y-value –5.
x = 0.5
B. x = 0.5
This is a vertical line
located at the x-value 0.5.
Holt McDougal Algebra 2
y = –5
2-3 Graphing Linear Functions
Example 6: Application
A ski lift carries skiers from an altitude of
1800 feet to an altitude of 3000 feet over a
horizontal distance of 2000 feet. Find the
average slope of this part of the mountain.
Graph the elevation against the distance.
Step 2
1 Graph
Find the
the
slope.
line.
The y-intercept
rise is 3000is– the
1800,
original
1800 ft.
or 1200 altitude,
ft.
Use (0, 1800) and (2000,
The run
2000
ft. on the
3000)
asistwo
points
line. Select a scale for each
The slope is
.
axis that will fit the data,
and graph the function.
Holt McDougal Algebra 2
2-3 Graphing Linear Functions
Check It Out! Example 6
A truck driver is at mile marker 624 on
Interstate 10. After 3 hours, the driver
reaches mile marker 432. Find his average
speed. Graph his location on I-10 in terms of
mile markers.
Step 1
2 Find
Graph
the
the
average
line. speed.
The
y-intercept
the
distance
= rate istime
distance traveled at 0
192 mi = rate  3 h
hours, 0 ft. Use (0, 0) and
(3, 192) as two points on
the line. Select a scale for
each axis that will fit the
data, and graph the
The
slope is 64 mi/h.
function.
Holt McDougal Algebra 2
2-3 Graphing Linear Functions
Lesson Quiz: Part 1
1. Determine whether the data could represent a
linear function.
x
–1
2
5
8
f(x)
–3
1
5
9
2. For 3x – 4y = 24, find the
intercepts, write in slopeintercept form, and graph.
x-intercept: 8;
y-intercept: –6;
y = 0.75x – 6
Holt McDougal Algebra 2
yes
2-3 Graphing Linear Functions
Lesson Quiz: Part 2
3. Determine if the line y = -3 is vertical or
horizontal. horizontal
4. The bottom edge of a roof is 62 ft above the
ground. If the roof rises to 125 ft above ground
over a horizontal distance of 7.5 yd, what is the
slope of the roof? 2.8
Holt McDougal Algebra 2