Download Lesson Plan -- Finding Slope

Lesson Plan -- Finding Slope
Chapter Resources
- Lesson 5-16 Slope
- Lesson 5-16 Slope Answers
- Lesson 5-18 Slope-Intercept Form
- Lesson 5-18 Slope-Intercept Form Answers
1
Name ———————————————————————
5-16
;TWXM
California
Standards
y
Words to Remember
Gr. 7 AF 3.3: Graph linear
functions, noting that
the vertical change
(change in y-value)
per unit of horizontal
change (change in
x-value) is always the
same and know that the
ratio (“rise over run”)
is called the slope of
the graph.
4
y2 2 y1
vertical change
rise
Slope: }} 5 }
5}
run
x
2 x1
horizontal change
2
Slope can be positive, negative,
zero, or undefined.
Zero
slope
23 22
Undefined
slope
LESSON
Date ————————————
Negative slope
3
1
O
4 x
1 2
22
Positive slope
24
Getting Started In this lesson you will use graphs of linear equations to
find the slopes of the lines.
Finding the Slope of a Line Using the
E@)584-
Rise
Ratio }
Run
Find the slope of the line through (3, 3) and (6, 5).
3FNFNCFS
The slope of a line is
the same between
any two points on
the line.
/PUJDF
A rise and a run can
be either positive or
negative.
;\MX Find the ratio of rise to
run between (3, 3) and (6, 5).
rise 2
}
run 5 }
3
;\MX Check the slope using
(3, 3) and (0, 1).
rise 22 2
5}
}
run 5 }
23 3
2
ANSWER The slope is }.
3
E@)584-
y
6
run 5 3
5
(6, 5)
4 rise 5 2
(3, 3)
3
rise 5 22
2
(0, 1)
run 5 23
O
1 2
3
4
5 6 x
22
Understanding the Slope of Horizontal
and Vertical Lines
Find the slope of y 5 3 and x 5 –2.
y
Solution
2
There is no vertical change between the points on the
line y 5 3. Therefore rise 5 0 5 0. The slope of y 5 3 is 0.
}
run
x 5 22
1
23
O
}
run
There is no horizontal change between the points on the
rise rise
. Since division by zero is not
line x 5 22. Therefore }
run 5 }
0
allowed, the slope of the line x 5 22 is undefined.
62
Math Intervention
Book 5 Algebraic Thinking
y53
1 x
Copyright © by McDougal Littell, a division of Houghton Mifflin Company.
Solution
Name ———————————————————————
T:A <0 1;
Fill in the missing information to find the slope of
the line.
1.
The slope of any
horizontal line is 0.
2.
y
24 23 22
y
(1, 1)
1
/PUJDF
Date ————————————
O
1 2
3
2
1
4 x
24 23 22
22
O
1 2
3
4 x
22
23
24
(21, 25) 25
rise
slope 5 }
The line has no
change. Therefore
______
5}5
2
The slope is
rise
}
}
run 5 run 5
.
.
.
The slope is
Slope Formula The slope of a line containing the points (x1, y1) and
y 2y
2
1
(x2, y2) can be found by using the formula slope 5 }
x 2x .
2
Copyright © by McDougal Littell, a division of Houghton Mifflin Company.
E@)584-
1
Using the Slope Formula
What is the slope of the line containing the points (5, 7) and (22, 4)?
3FNFNCFS
When calculating
the slope from the
coordinates of two
points, subtract
the x-values in the
same order that you
subtract the
y-values.
Solution
y 2y
2
1
slope 5 }
x 2x
2
427
5}
22 2 5
23
5}
27
3
5}
7
T:A <0 1;
/PUJDF
The slope of any
vertical line is
undefined.
Slope formula
1
Substitute values for y2, y1, x2, and x1.
Simplify.
Simplify.
Find the slope of the line containing the given points.
3. (28, 2) and (6, 9)
y2 2 y1
slope 5 }
x2 2 x1
5
92
2 28
5}5
The slope is
4. (8, 2) and (8, 23)
y 2y
2
1
slope 5 }
x 2x
2
5
1
22
82
5}
.
The slope is
.
Math Intervention
Book 5 Algebraic Thinking
63
Name ———————————————————————
Date ————————————
Summarize
Finding the Slope of a Line using Rise over Run
You can find the slope of a line by counting the rise (vertical change)
and run (horizontal change) between two points on the graph. Then
rise
write the fraction for }
run and simplify.
Finding the Slope of a Line Using the Slope Formula
y 2y
2
1
The slope can also be found by calculating x}
2 x . A horizontal line has
2
1
0 slope. A vertical line has undefined slope. A line sloping up to the right
has a positive slope. A line sloping down to the right has a negative slope.
Practice
Tell whether the slope of the line is positive, negative, zero, or undefined.
1.
2.
y
1
22
3.
y
22
1 2 x
O
O
4.
y
2
1
y
2
2
1
22
1 2 x
22
O
22
x
1
O
1 2 x
22
22
y
5.
6.
1
22
O
5
6
4
(26, 0)
2
1 2 x
22
slope 5
y
(26, 6)
28
______
24
O
x
The line has no
run
____________ change.
rise rise
So }
run 5 }
1
Division by
The slope is
is
undefined. Therefore,
.
the slope is ____________ .
Find the slope of the line.
y
7.
(21.5, 0)
1
22
O
8.
y
8
6
1 2 x
(21, 22)
Math Intervention
Book 5 Algebraic Thinking
4
2
1
O
64
9.
y
4
1
2 3 x
24
(1, 7)
(0, 3)
2 4 x
Copyright © by McDougal Littell, a division of Houghton Mifflin Company.
Fill in the missing information to find the slope of the line.
Name ———————————————————————
Date ————————————
Find the slope of the line containing the given points.
10. (0, 5) and ( 21, 4)
11. (3, 6) and (8, 6)
y 2y
y 2y
2
1
slope 5 }
x 2x
2
5
2
1
slope 5 }
x 2x
1
2
42
20
26
32
5}
5}
5
5
The slope is
.
The slope is
.
12. (2, 5) and (2, 21)
13. (211, 29) and (28, 23)
14. (212, 22) and (29, 13)
15. (22, 26) and (8, 212)
1
1 1
3 2
2
1
1 2
2 3
16. 2}, } and 2}, }
Copyright © by McDougal Littell, a division of Houghton Mifflin Company.
5
1
2
17.
1 2}25, 2}34 2 and 1 }35, }14 2
,1,A7=/ -<1<'
18. Fill in the missing words. The slope of a line can be found
by writing the fraction ____________ /____________.
19. Find the error. Lihue found the slope of the line containing
2
(5, 6) and (3, 9) to be 2}. What was his mistake?
3
__________________________________________________________________________
__________________________________________________________________________
__________________________________________________________________________
20. Explain your reasoning. A line contains (2, 4) and (23, 5).
524
. Your friend found the slope
You found the slope using }
2
322
425
using }. Who is correct? Explain.
2 2 (23)
__________________________________________________________________________
__________________________________________________________________________
__________________________________________________________________________
Math Intervention
Book 5 Algebraic Thinking
65
Answer Key
Lesson 5-16, pp. 62–65
Try this:
1. run; 6; 3; 3
2. vertical; 0; 0; 0
7
7
3. 2; 6; 7; 222; 2​}
22 ​; 2​}
22 ​ 21
4. 23; 8; }
​ 0 ​; undefined
Practice:
1. zero
2. positive
3. undefined
4. negative
5. rise; 2; 1; 2
6. horizontal; 0; 0; undefined
7. 24
8. 0
9. 4
10. 5; –1; –1; –1; 1; 1
11. 6; 8; 0; –5; 0; 0
12. undefined
13. 2
14. 5
3
15. 2​}
5 ​ 16. 21
17. 1
18. rise; run
19. Sample answer: He put the difference of the x-values in the
numerator and the difference of the y-values in the denominator.
He should have put the difference of the y-values in the numerator
and the difference of the x-values in the denominator.
20. Sample answer: You are both correct. The order you subtract the
coordinates of the points does not matter as long as you are consistent
and subtract the coordinates in the numerator and denominator in the same order.
Name ———————————————————————
5-18
;TWXM1V\MZKMX\.WZU
California
Standards
Words to Remember
Gr. 7 AF 3.0: Students
graph and interpret
linear and some
nonlinear functions.
Gr. 7 AF 3.3: Graph
linear functions, noting
that the vertical change
(change in y-value)
per unit of horizontal
change (change in
x-value) is always the
same and know that the
ratio (“rise over run”)
is called the slope of
the graph.
y-intercept: The y-value of the point where a line crosses the y-axis
Slope-intercept form: The equation of a line in the form y 5 mx 1 b,
where m is the slope of the line and b is the y-intercept
Getting Started In Lesson 5-15 you learned how to graph a linear
equation using a table of values. In this lesson you will learn another way to
graph a linear equation.
E@)584-
Remember
The slope m is the
Finding the Slope and y-Intercept
Find the slope and y-intercept of the line y 5 3x 1 5.
Solution
Compare the equation to y 5 mx 1 b.
coefficient of x.
y 5 mx 1 b
m is the slope; b is the y-intercept.
y 5 3x 1 5
In this equation, m 5 3 and b 5 5.
ANSWER The slope m is 3. The y-intercept b is 5.
T:A <0 1;
Fill in the missing information to solve the problem.
3
4
1. Find the slope and y-intercept of the line y 5 }x 2 2.
Remember
The equation
y 5 }3x 2 2 can be
4
rewritten as
y 5 }3 x 1 (22).
3
Compare y 5 }x 2 2 to y 5 mx 1
4
y 5 mx 1
y5
x
.
m is the slope;
22
is the y-intercept.
3
In the equation y 5 }
x – 2, m 5
4
4
The slope m is
70
Math Intervention
Book 5 Algebraic Thinking
. The y-intercept b is
.
, and b 5
.
Copyright © by McDougal Littell, a division of Houghton Mifflin Company.
LESSON
Date ————————————
Name ———————————————————————
E@)584-
Date ————————————
Graphing a Line in Slope-Intercept Form
Graph the line y 5 4x 1 3.
Solution
;\MX Identify the slope and y-intercept.
Compare y 5 4x 1 3 to y 5 mx 1 b.
This shows you that the slope m
is 4 and the y-intercept b is 3.
y
7
(1, 7)
6
run 5 1
rise 5 4 5
4
(0, 3)
3
;\MX Plot the y-intercept at (0, 3).
Remember
Write the slope as the
4
fraction } to identify
1
the rise and the run.
2
1
;\MX Use the slope to graph a second point.
4
rise }
slope 5 }
run 5 1
From (0, 3) move 4 units up and
24 23 22
3
4 x
1 2
3
4 x
1 2
3
4 x
7
6
5
4
3
Graph this point.
;\MX Draw the graph of the
line between the two points.
Don’t Forget You will need to move down from
the y-intercept if the slope is negative.
Copyright © by McDougal Littell, a division of Houghton Mifflin Company.
1 2
y
1 unit right to (1, 7 ).
T:A <0 1;
O
24 23 22
O
Fill in the missing information.
1
2
y
2. Follow these steps to graph y 5 2}x 2 1.
4
;\MX Identify the slope and
.
1
1
Rewrite y 5 2}x 2 1 as y 5 2}x
2 1.
2
2
1
to y 5 mx 1 b.
Compare y 5 2}x
2
This shows you that the slope m is
and the y-intercept is
;\MX Plot the y-intercept at (
3
2
1
24 23 22
O
22
23
24
.
,
).
;\MX Use the slope to graph a second point.
_____
slope 5 }
.
run 5 }
2
unit down and
From b move
This takes you to the point (
,
units right.
). Graph this point.
1
;\MX Draw the graph of y 5 2}x 2 1 between (
2
y-intercept.
,
) and the
Math Intervention
Book 5 Algebraic Thinking
71
Name ———————————————————————
Date ————————————
Summarize
Finding the Slope and y-Intercept
When an equation is in the form y 5 mx 1 b, the value of m is the
slope and the value of b is the y-intercept.
Graphing a Line Using the Slope and y-Intercept
To graph a line in slope-intercept form, first plot the y-intercept
on the y-axis. Then use the rise and the run to move from the
y-intercept to a second point on the line. Graph this point and draw
a line through that point and the y-intercept.
Practice
Find the slope and y-intercept of the line.
1. y 5 8x 1 2
Compare y 5 8x 1 2 to y = mx 1
y5
m is the slope;
In the equation y 5 8x 1 2, m 5
2.
x
The slope m is
is the y-intercept.
. The y-intercept b is
.
.
1
5
2. y 5 }x 2 3 __________________
2
3
and b 5
3. y 5 27x 2 4 ___________
1
4. y 5 2}x 1 } _________________
Copyright © by McDougal Littell, a division of Houghton Mifflin Company.
y 5 mx 1
.
5. y 5 x 2 10___________
2
6. Explain how you would use the slope and y-intercept to graph
y 5 24x 1 3.
The y-intercept is
, so graph (
,
).
rise
5 }, so from the y-intercept, move
The slope 5 }
___
1
___________ then 3 units ___________ .
Draw a line through this point and (
,
units
).
Graph the line using the slope and y-intercept.
7. y 5 3x 1 1
4
24 23 22
Math Intervention
Book 5 Algebraic Thinking
y
4
3
2
3
2
1
1
O
22
23
24
72
1
2
8. y 5 }x 2 2
y
1 2
3
4 x
24 23 22
O
22
23
24
1 2
3
4 x
Name ———————————————————————
Date ————————————
Graph the line using the slope and y-intercept.
9. y 5 2x 2 6
10. y 5 4x – 6
y
8
28 26 24
y
8
6
4
6
4
2
2
O
2 4
6
28 26 24
8 x
24
26
28
11. y 5 x
12. y 5 x 1 3
y
3
5
3
2
2
1
O
2 4
6
24 23 22
8 x
Copyright © by McDougal Littell, a division of Houghton Mifflin Company.
O
1 2
3
4 x
1 2
3
4 x
22
23
24
2
3
14. y 5 2}x 1 4
4
y
4
3
2
3
2
1
1
O
8 x
y
6
4
y
24 23 22
6
4
24
26
28
13. y 5 }x 1 2
2 4
24
26
28
8
28 26 24
O
1 2
3
4 x
22
23
24
24 23 22
O
22
23
24
,1,A7=/ -<1<'
15. Fill in the missing words. In the equation y 5 2x 1 6, 2 is
the ____________ and 6 is the ____________. This equation is in
_______________________form.
16. Explain your reasoning. Do you think it would be easier to
2 x 1 1.4 using a table of values or the slope
graph the line y 5 }
3
and y-intercept? Explain.
__________________________________________________________________________
__________________________________________________________________________
__________________________________________________________________________
Math Intervention
Book 5 Algebraic Thinking
73
Answer Key
Lesson 5-18, pp. 70–73
Try this:
3
3
3
1. b; b; b; }
​ 4 ​; +; }
​ 4 ​; –2; }
​ 4 ​; –2
1
2. y-intercept; 1; 1; 21; 2​}
2 ​; 21; 0;
21; rise; 21; 1; 2; 2; 22; 2; 22;
y
4
3
2
1
1 2
24 23 22
3
4 x
22
23
24
Practice:
1. b; b; b; 8; 1; 8; 2; 8; 2
1
2. slope 5 ​ } ​, y-intercept 5 23
5
3. slope 5 27, y-intercept 5 24
2
1
4. slope 5 2​ } ​, y-intercept 5 }
​ 2 ​ 3
5. slope 5 1, y-intercept 5 210
6. 3; 0; 3; run; 24; 4; down; right; 0; 3
7.
4
3
2
1
y
1 2
24 23 22 21
3
4 x
23
24
8.
4
3
2
1
24 23 22
O
23
24
y
1 2
4 x
Answer Key
9.
8
6
4
2
y
2 4
O
26
6 8 x
28
10.
8
6
4
2
y
O
28 26 24
2 4
6 8 x
2 4
6 8 x
1 2
3 4 x
24
26
11.
8
6
4
2
y
28 26 24 22
24
26
28
12.
4
3
2
1
24
22
O
22
23
24
y
Answer Key
13.
4
3
y
1
23 22
O
1 2
3 4 x
1
3 4 x
22
23
24
y
14.
3
2
1
24 23 22
O
2
22
23
24
15. slope; y-intercept; slope-intercept
16. Sample answer: It would be.ctions and decimals.