warm up solve each equation for y. 1. 7x + 2y = 6 2. 3. if 3x = 4y + 12, find y when x = 0. 4. if a...
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![Page 1: Warm Up Solve each equation for y. 1. 7x + 2y = 6 2. 3. If 3x = 4y + 12, find y when x = 0. 4. If a line passes through (–5, 0) and (0, 2), then it passes](https://reader035.vdocument.in/reader035/viewer/2022062417/55155d9f55034674578b4887/html5/thumbnails/1.jpg)
Warm UpSolve each equation for y.
1. 7x + 2y = 6
2.
3. If 3x = 4y + 12, find y when x = 0.
4. If a line passes through (–5, 0) and (0, 2), then it passes through all but which quadrant.
y = –2x – 8
y = –3
IV
![Page 2: Warm Up Solve each equation for y. 1. 7x + 2y = 6 2. 3. If 3x = 4y + 12, find y when x = 0. 4. If a line passes through (–5, 0) and (0, 2), then it passes](https://reader035.vdocument.in/reader035/viewer/2022062417/55155d9f55034674578b4887/html5/thumbnails/2.jpg)
I can determine whether a function is linear.
I can graph a linear function given two points, a table, an equation, or a point and a slope.
2.3 Linear Functions and Slope Intercept Form
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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.
Time (h) 0 1 2 3 4
Distance from Land (mi) 350 325 300 275 250
+1
–25
+1
–25
+1
–25
+1
–25
This rate can be expressed as .
Notice that the rate of change is constant. The hurricane moves 25 miles closer each hour.
![Page 4: Warm Up Solve each equation for y. 1. 7x + 2y = 6 2. 3. If 3x = 4y + 12, find y when x = 0. 4. If a line passes through (–5, 0) and (0, 2), then it passes](https://reader035.vdocument.in/reader035/viewer/2022062417/55155d9f55034674578b4887/html5/thumbnails/4.jpg)
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).
![Page 5: Warm Up Solve each equation for y. 1. 7x + 2y = 6 2. 3. If 3x = 4y + 12, find y when x = 0. 4. If a line passes through (–5, 0) and (0, 2), then it passes](https://reader035.vdocument.in/reader035/viewer/2022062417/55155d9f55034674578b4887/html5/thumbnails/5.jpg)
Review of Formulas
Formula for Slope m y2 y1x2 x1
Ax By C Standard Form
y y1 m x x1 Point-Slope Form
ymx bSlope-intercept Form
*where A>0 and A, B, C are integers
![Page 6: Warm Up Solve each equation for y. 1. 7x + 2y = 6 2. 3. If 3x = 4y + 12, find y when x = 0. 4. If a line passes through (–5, 0) and (0, 2), then it passes](https://reader035.vdocument.in/reader035/viewer/2022062417/55155d9f55034674578b4887/html5/thumbnails/6.jpg)
Determine whether the data set could represent a linear function.
Example 1A: Recognizing Linear Functions
x –2 0 2 4
f(x) 2 1 0 –1
+2
–1
+2
–1
+2
–1
The rate of change, , is constant . So
the data set is linear.
![Page 7: Warm Up Solve each equation for y. 1. 7x + 2y = 6 2. 3. If 3x = 4y + 12, find y when x = 0. 4. If a line passes through (–5, 0) and (0, 2), then it passes](https://reader035.vdocument.in/reader035/viewer/2022062417/55155d9f55034674578b4887/html5/thumbnails/7.jpg)
Determine whether the data set could represent a linear function.
Example 1B: Recognizing Linear Functions
x 2 3 4 5
f(x) 2 4 8 16
+1
+2
+1
+4
+1
+8
The rate of change, , is not constant.
2 ≠ 4 ≠ 8. So the data set is not linear.
![Page 8: Warm Up Solve each equation for y. 1. 7x + 2y = 6 2. 3. If 3x = 4y + 12, find y when x = 0. 4. If a line passes through (–5, 0) and (0, 2), then it passes](https://reader035.vdocument.in/reader035/viewer/2022062417/55155d9f55034674578b4887/html5/thumbnails/8.jpg)
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.
![Page 9: Warm Up Solve each equation for y. 1. 7x + 2y = 6 2. 3. If 3x = 4y + 12, find y when x = 0. 4. If a line passes through (–5, 0) and (0, 2), then it passes](https://reader035.vdocument.in/reader035/viewer/2022062417/55155d9f55034674578b4887/html5/thumbnails/9.jpg)
Example 2A: Graphing Lines Using Slope and a Point
Plot the point (–1, –3).
Graph the line with slope that passes through (–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.
![Page 10: Warm Up Solve each equation for y. 1. 7x + 2y = 6 2. 3. If 3x = 4y + 12, find y when x = 0. 4. If a line passes through (–5, 0) and (0, 2), then it passes](https://reader035.vdocument.in/reader035/viewer/2022062417/55155d9f55034674578b4887/html5/thumbnails/10.jpg)
Example 2B: Graphing Lines Using Slope and a Point
Plot the point (0, 2).
Graph the line with slope that passes through (0, 2).
You can move down 3 units and right 4 units, or move up 3 units and left 4 units.
The negative slope can be
viewed as
![Page 11: Warm Up Solve each equation for y. 1. 7x + 2y = 6 2. 3. If 3x = 4y + 12, find y when x = 0. 4. If a line passes through (–5, 0) and (0, 2), then it passes](https://reader035.vdocument.in/reader035/viewer/2022062417/55155d9f55034674578b4887/html5/thumbnails/11.jpg)
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.
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Example 3: Graphing Lines Using the Intercepts
Find the x-intercept: 4x – 2y = 16
Find the intercepts of 4x – 2y = 16, and graph the line.
4x – 2(0) = 16
Find the y-intercept: 4x – 2y = 16
4x = 16
x = 4
4(0) – 2y = 16
–2y = 16
y = –8
Substitute 0 for y.
The x-intercept is 4.
Substitute 0 for x.
The y-intercept is –8.
x-intercept
y-intercept
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Check It Out! Example 3
Find the x-intercept: 6x – 2y = –24
Find the intercepts of 6x – 2y = –24, and graph the line.
6x – 2(0) = –24
Find the y-intercept: 6x – 2y = –24
6x = –24
x = –4
6(0) – 2y = –24
–2y = –24
y = 12
Substitute 0 for y.
The x-intercept is –4.
Substitute 0 for x.
The y-intercept is 12.
x-intercept
y-intercept
![Page 14: Warm Up Solve each equation for y. 1. 7x + 2y = 6 2. 3. If 3x = 4y + 12, find y when x = 0. 4. If a line passes through (–5, 0) and (0, 2), then it passes](https://reader035.vdocument.in/reader035/viewer/2022062417/55155d9f55034674578b4887/html5/thumbnails/14.jpg)
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.
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Example 4A: Graphing Functions in Slope-Intercept Form
Solve for y first.
Write the function –4x + y = –1 in slope-intercept form. Then graph the function.
–4x + y = –1
y = 4x – 1
Add 4x to both sides.+4x +4x
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.
![Page 16: Warm Up Solve each equation for y. 1. 7x + 2y = 6 2. 3. If 3x = 4y + 12, find y when x = 0. 4. If a line passes through (–5, 0) and (0, 2), then it passes](https://reader035.vdocument.in/reader035/viewer/2022062417/55155d9f55034674578b4887/html5/thumbnails/16.jpg)
Example 4B: Graphing Functions in Slope-Intercept Form
Solve for y first.
Write the function in slope-intercept
form. Then graph the function.
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.
Multiply both sides by
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Check It Out! Example 4A
Solve for y first.
Write the function 2x – y = 9 in slope-intercept form. Then graph the function.
2x – y = 9
–y = –2x + 9
Add –2x to both sides. –2x –2x
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.
![Page 18: Warm Up Solve each equation for y. 1. 7x + 2y = 6 2. 3. If 3x = 4y + 12, find y when x = 0. 4. If a line passes through (–5, 0) and (0, 2), then it passes](https://reader035.vdocument.in/reader035/viewer/2022062417/55155d9f55034674578b4887/html5/thumbnails/18.jpg)
Check It Out! Example 4B
Solve for y first.
Write the function 5x = 15y + 30 in slope-intercept form. Then graph the function.
5x = 15y + 30
5x – 30 = 15y
Subtract 30 from both sides. –30 –30
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.
![Page 19: Warm Up Solve each equation for y. 1. 7x + 2y = 6 2. 3. If 3x = 4y + 12, find y when x = 0. 4. If a line passes through (–5, 0) and (0, 2), then it passes](https://reader035.vdocument.in/reader035/viewer/2022062417/55155d9f55034674578b4887/html5/thumbnails/19.jpg)
An equation with only one variable can be represented by either a vertical or a horizontal line.
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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.
The slope of a vertical line is undefined.
The slope of a horizontal line is zero.
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Example 5: Graphing Vertical and Horizontal Lines
Determine if each line is vertical or horizontal.
A. x = 2
B. y = –4
This is a vertical line located at the x-value 2. (Note that it is not a function.)
This is a horizontal line located at the y-value –4.
x = 2
y = –4
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Check It Out! Example 5
Determine if each line is vertical or horizontal.
A. y = –5
B. x = 0.5
This is a horizontal line located at the y-value –5.
This is a vertical line located at the x-value 0.5.
x = 0.5
y = –5
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Assignment #10
pg 78-79 (10pts) #9-36 x3
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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 1 Find the slope.
The rise is 3000 – 1800, or 1200 ft.
The run is 2000 ft.
The slope is .
Step 2 Graph the line.The y-intercept is the original altitude, 1800 ft. Use (0, 1800) and (2000, 3000) as two points on the line. Select a scale for each axis that will fit the data, and graph the function.
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Check It Out! Example 6A 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 Find the average speed.
distance = rate time
The slope is 64 mi/h.
192 mi = rate 3 h
The y-intercept is the distance traveled at 0 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 function.
Step 2 Graph the line.
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Lesson Quiz: Part 1
1. Determine whether the data could represent a linear function.
yes
x-intercept: 8; y-intercept: –6; y = 0.75x – 6
x –1 2 5 8
f(x) –3 1 5 9
2. For 3x – 4y = 24, find the intercepts, write in slope-intercept form, and graph.
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Lesson Quiz: Part 2
3. Determine if the line y = -3 is vertical or horizontal. horizontal
2.8
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?