name date ms - white plains middle school · 1 chapter 5 - 2 day 1 - using intercepts to graph...
TRANSCRIPT
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Algebra
Chapter 5: LINEAR FUNCTIONS
Name:______________________________
Teacher:____________________________
Pd: _______
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Table of Contents
Chapter 5-2: x- and y- Intercepts SWBAT: Use intercepts to graph lines
Pgs. #1 - 4
Hw pgs. #5 – 6
Chapter 5-3/5-4: Slope SWBAT: Calculate the slope from any two points
Pgs. #7 - 11
Hw pgs. #12 – 13
Chapter 5-5: Direct Variation SWBAT: Identify, write, and graph direct variations
Pgs. #14 - 18
Hw pgs. #18 – 19
Chapter 5-6: Slope – Intercept Form SWBAT: Write and Graph a linear equation in Slope – Intercept Form
Pgs. #20 – 24
Hw pgs. #25 – 27
Chapter 5-7: Point - Slope Form SWBAT: Write and Graph a linear equation in Point - Slope Form
Pgs. #28 – 32
Hw pgs. #33 – 34
Review: Sections 5-2 through 5 - 7
Pgs. #35 – 38
Chapter 5-8: Slope of Parallel and Perpendicular Lines SWBAT: Calculate the slope of Parallel and Perpendicular Lines
Pgs. #39 – 42
Hw pgs. #43 – 44
Chapter 5-8: Equations of Parallel and Perpendicular Lines SWBAT: Write the equation of Parallel and Perpendicular Lines
Pgs. #45 – 48
Hw pgs. #49 – 50
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1
Chapter 5 - 2
Day 1 - Using Intercepts to Graph Lines
Warm Up
Solve each equation for the variable.
1) 5x + 0 = –10 2) 33 = 0 + 3y
Example 1: Find the x-intercept and y-intercept from the graph.
1. 2.
x-intercept: _____________ x-intercept: _____________
y-intercept: _____________ y-intercept: _____________
Definitions:
The y-intercept is the y-coordinate of the point where the graph intersects the y-axis.
The x-intercept is the x-coordinate of the point where the graph intersects the x-axis.
y
x x
y
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2
Practice: Find the x-intercept and y-intercept from the graph.
3. gf 4.
x-intercept: _____________ x-intercept: _____________
y-intercept: _____________ y-intercept: _____________
Example 2: Find the x-intercept and y-intercept from the equation and then graph.
5. 5x – 2y = 10
x- intercept
To find the x-intercept:
replace y with 0 and solve for x.
y-intercept
To find the y-intercept:
replace x with 0 and solve for y
5x – 2y = 10
The x-intercept is __________
5x – 2y = 10
The y-intercept is __________
6. –10x + 6y = 30
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Practice: Find the x-intercept and y-intercept from the equation and then graph.
7. 4x + 2y = 8
8. 3x – 7y = 21
9.
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Challenge
SUMMARY
Exit Ticket
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5
Day1 – Homework
Using Intercepts to Graph Lines
Find the x-intercept and y-intercept for the graphs below.
1) 2)
x-intercept: _______________ x-intercept: _______________
y-intercept: _______________ y-intercept: _______________
3) 4)
x-intercept: _______________ x-intercept: _______________
y-intercept: _______________ y-intercept: _______________
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Determine the x-intercept and y-intercept for each equation and graph.
5) 3x + 2y = –6 6) x – 4y = 4
7) y = 0.25x – 1 8) 2x + 3y = 6
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Chapter 5 – 3/5 - 4
Day 2 – Slope
Warm - Up
Find the x- and y- intercepts.
a. b.
x-intercept: _______________ x-intercept: _______________
y-intercept: _______________ y-intercept: _______________
Motivation:
The slope of a line measures the steepness of the line.
We’re familiar with the word slope as it relates to mountains.
Skiers and snowboarders refer to “hitting the slopes.”
Slope measures the ratio of the change in the y-value of a line to a
given change in its x-value.
Finding Slope
DEFINITION OF SLOPE
2 1
2 1
rise change in slope
run change in
Y Y
X X
y ym
x x
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Example 1: Calculating the slope from a Graph
a: Find the slope of the line. b: Find the slope of the line.
1st Point: _________ 2
nd Point: ________ 1
st Point: _________ 2
nd Point: _________
2 1
2 1
slopeY Y
X X
ym
x
2 1
2 1
slopeY Y
X X
ym
x
c: Find the slope of the line. d: Find the slope of the line.
1st Point: _________ 2
nd Point: _________ 1
st Point: ________ 2
nd Point: _________
2 1
2 1
slopeY Y
X X
ym
x
2 1
2 1
slopeY Y
X X
ym
x
x
y
x
y
x
y
x
y
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e. f.
Example 2: Calculating the slope from a set up points
a) Find the slope of the line that passes through the points 8, 7 and 4, 5 .
2 1
2 1
Y Y
X Xm
1,1
)7,8(yx
and
2,2
)5,4(yx
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1,1
)3,4(yx
and
2,2
)2,5(
yx
b) Find the slope of the line that passes through the points (4, 3) and (–5, –2).
2 1
2 1
Y Y
X Xm
c) Find the slope of the line that passes through the points (–4, –7) and (–3, –5).
2 1
2 1
Y Y
X Xm
Example 3: Calculating the Slope from a Table of Values
a) Find the slope of the line that contains the points from the table
b) Find the slope of the line that contains the points from the table
1,1
)7,4(
yx
and 2,2
)5,3(
yx
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How to remember slope?
Ski - Slope House
Challenge
Summary
Exit Ticket
1. Fill in the blanks.
2.
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Day2 – Homework
Slope
1) Find the slope of the line that passes through each of the following sets of points.
(a) (b) (c)
(d) (e) (f)
2) Find the slope of each of the following lines graphically:
a) fkdjflkdjfjkd b)
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c) d)
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Chapter 5 - 5
Day 3 – Direct Variation
Warm Up
Find the slope of the linear function represented by the table below.
1)
2)
Motivation:
Who uses this?
A recipe for paella calls for 1 cup of rice to make 5 servings. In other words, a chef
needs 1 cup of rice for every 5 servings.
The equation y = 5x describes this relationship. In this relationship, the number of
servings varies directly with the number of cups of rice.
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Direct Variation
Example1: Identifying Direct Variations from an Equation
Tell whether the equation represents a direct variation. If so, identify the constant of
variation.
1) y = 21x 2) 3x + y = 8
Direct Variation? _____________ Direct Variation? ____________
Constant of Variation: ___________ Constant of Variation: __________
3) –4x + 3y = 0 4) 3x + 0 = 15y
Direct Variation? _____________ Direct Variation? ____________
Constant of Variation: ___________ Constant of Variation: __________
What happens if you solve y = kx for k?
Let’s see…
In a direct variation, the ratio ________
is equal to the constant of variation.
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Example 2: Identifying Direct Variations from Ordered Pairs
Tell whether the relationship is a direct variation. Explain.
5) 6)
Find y
x for each ordered pair. Find
y
x for each ordered pair.
Direct Variation? _____________ Direct Variation? ____________
Constant of Variation: ___________ Constant of Variation: __________
Example 3: Identifying Direct Variations from a Graph
Tell whether the relationship is a direct variation. Explain.
7) 8)
Direct Variation? _____________ Direct Variation? ____________
Constant of Variation: ___________ Constant of Variation: __________
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Example 4: Writing and Solving Direct Variation Equations
9) The value of y varies directly with x, and y = 3 when x = 9. Find y when x = 21.
Use a proportion: 9
3
x
y
10) The value of y varies directly with x, and y = 4.5 when x = 0.5. Find y when x = 10.
Challenge
SUMMARY
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Exit Ticket
Day 3 – Homework
Direct Variation
Tell whether each equation is a direct variation. If so, identify the constant of variation.
1) y = 3x 3) y = 2x – 9
2) 2x + 3y = 0 4) 3y = 9x
Find the value of y
xfor each ordered pair. Then tell whether the relationship is a direct variation.
5) 6) 7)
________________ _________________ _________________
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Tell whether each graph is a direct variation. If so, identify the constant of variation.
8) 9)
_______________ ________________
10) The value y varies directly with x, and y = –18 when x = 6. Find y when x = –8.
11) The value of y varies directly with x and y = ½ when x = 5. Find y when x = 30.
12)
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Chapter 5 – 6
Day 4 - Slope – Intercept Form
Warm Up
Motivation:
Who uses this?
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Graphing by using slope and y-intercept
1. 24
3 xy m = ______ 2. 42 xy m = ______
y-intercept = b = (0, ____ ) y-intercept = b = (0, ____ )
Practice
Graphing by using slope and y-intercept
3. 32
1 xy m = ______ 4. xy 3 m = ______
y-intercept = b = (0, ____ ) y-intercept = b = (0, ____ )
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5. y = –4
Make a Table of Values:
x y
What do you notice about the y values? __________
Graph and label the lines 5 and 2y y on the same grid.
6. x = 2
Make a Table of Values:
x y
What do you notice about the x values? __________
Graph and label the lines x = –1 and x = 5 on the same grid.
EQUATIONS OF HORIZONTAL LINES
where (or simply ) 0 y by mx b m
y
x
EQUATIONS OF VERTICAL LINES
where is the of the lineintercepta xx a
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7.
Writing Equations of Graphs
11. Write the equation in slope–intercept form. Then graph the line described by the equation.
3x + 2y = 5
8. 9. 10.
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12. Write the equation in slope–intercept form. Then graph the line described by the equation.
4x - 2y = 14
Challenge
SUMMARY
Exit Ticket
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Day 4 – Homework
Slope – Intercept Form
1) Which of the following lines has a slope of 5 and a y-intercept of 3 ?
(1) 5 3y x (3) 3 5y x
(2) 5
3y x (4) 3 5y x
2) Which of the following equations represents the graph shown?
(1) 3
32
y x (3) 2
33
y x
(2) 3
22
y x (4) 2
23
y x
3) Graph each line.
(a) y = –3x+ 2 (b) y = 1
2 x+ 0
y
x
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(c) y = 6x + 3 (d) y = 2
3x + 9
(e) y = –1 (f) x = –5
(g) y = 8 (h) x = 7
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3) Write an equation of each line in slope – intercept form. Identify the slope and y-intercept.
a. 2 2 4x y b. 5 7x y
c. 6x + 2y = 8 d. 10 5 15y
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Day 5 – Point Slope Form
Warm – Up
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Yesterday, we learned how to graph equations using the slope and the y-intercept. Today we are
going to write equations of lines.
First, let’s see how to use the equation.
Example 1: Graph the linear equation.
y – 1 = 2(x – 3) m = _______ pt = ( ___, ____ ) Practice: Graph the linear equation.
y + 4 = -¼(x – 8) m = _______ pt = ( ___, ____ )
Equations of Lines
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Example 2: Converting Point-Slope to Slope-Intercept Form and Standard Form.
Write in Slope-Intercept form and Standard Form.
Practice: Converting Point-Slope form to Slope intercept Form
Writing Equations of Lines
Example 3: Write the equation of a line given the slope and a point.
(-3, -4); m = –3
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Practice: Write the equation of a line given the slope and a point.
(1, 2); m = –3
Example 4: Write the equation of a line passing through the two points given.
(10, 20) and (20, 65)
Step 1:
Step 2: plug m, and point into equation.
y – y1 = m(x – x1)
Practice: Write the equation of a line passing through the two points given.
(2, –5) and (–8, 5)
Step 1:
Step 2: plug m, and point into equation.
y – y1 = m(x – x1)
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Challenge
Write the equation of a line in point-slope form passing through the two points given.
(f, g) (h, j)
SUMMARY
Exit Ticket
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Day 5 – Point Slope Form - HW
1) Which equation describes the line through (–5, 1) with the slope of 1?
(a) y = x – 6 (c) y = –5x + 6
(b) y = –5x – 6 (d) y = x + 6
2) A line contains (4, 4) and (5, 2). What is the slope and y – intercept?
(a) slope = –2; y – intercept = 2 (c) slope = –2; y – intercept = 12
(b) slope = 1.2; y – intercept = –2 (d) slope = 12; y – intercept = 1.2
Write an equation for the line with the given slope and point in slope-intercept form.
3) slope = 3; (–4, 2)
Equation: ______________________
4) slope = –1; (6, –1 )
Equation: ______________________
5) slope = 0; (1, –8)
Equation: ______________________
6) slope = –9; (–2, –3)
Equation: ______________________
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Write an equation for the line through the two points in slope intercept form.
7) (2, 1); (0, –7)
Equation: ______________________
8) (–6, –6); (2, –2)
Equation: ______________________
9) (–2, –3); (–1, –4)
Equation: ______________________
10) (6, 12); (0, 0)
Equation: ______________________
Write an equation for the line for each graph.
11) 12)
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Day 6 – Review of Sections 5-2 through 5-6
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Write the equation of the lines below. Write your answer in slope-intercept form.
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Day 7 – Slopes of Parallel and Perpendicular Lines (5 – 8)
Warm Up
Directions: Find the reciprocal.
1) 2 2) 3
1 3)
9
5
Directions: Find the slope of the line that passes through each pair of points.
4) (2, 2) and (–1, 3) 5) (3, 4) and (4, 6) 6) (5, 1) and (0, 0)
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m m
m =
m =
m =
m =
m =
m =
m =
m =
m =
m =
m =
m =
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Challenge
If ||PQ RS and the slope of 1
4
xPQ
and the slope of RS is
6
8, then find the value of x. Justify
algebraically or numerically.
SUMMARY
Exit Ticket
1.
2.
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Day 7 – Slopes of Parallel and Perpendicular Lines HW
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7)
8)
9)
10)
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Day 8 – Equations of Parallel and Perpendicular Lines (5 – 8)
Warm Up
Determine the equation of the line that passes through the two points 2,4 and 8,0 .
1) Write an equation for the line that passes through (4, 10) and is parallel to the line described by y = 3x + 8.
Step 1: m = _____
Step 2: plug in the point into y – y1 = m(x – x1) and solve for y.
Equation: __________________________________
2) Write an equation for the line that passes through (–2 , 5) and is parallel to the line described by y = 2
1x – 7.
Step 1: m = _____
Step 2: plug in the point into y – y1 = m(x – x1) and solve for y.
Equation: __________________________________
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3) Write an equation for the line that passes through (2, –1) and is perpendicular to the line described by
y = 2x – 5.
Step 1: m = _____
Step 2: plug in the point into y – y1 = m(x – x1) and solve for y.
Equation: __________________________________
4) Write an equation for the line that passes through (2, 6) and is perpendicular to the line described by
y = -3
1x + 2.
Step 1: m = _____
Step 2: plug in the point into y – y1 = m(x – x1) and solve for y.
Equation: __________________________________
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Regents Practice
5) What is the slope of a line parallel to the line whose equation is y = -4x + 5?
6) What is the slope of a line parallel to the line whose equation is 3x + 6y = 6?
7)
8) Kjk
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Challenge
SUMMARY
Exit Ticket
1.
2.
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Day 8 – Equations of Parallel and Perpendicular Lines - HW
1) Which equation represents a line parallel to the
x-axis?
(a) y 5 (c) x 3
(b) y x 5 (d) x y 3
2) Which equation represents a line parallel to
the line y = 2x – 5?
(a) y = 2x + 5 (c) y = 5x – 2
(b) y = – x – 5 (d) y = –2x – 5
3) Which equation represents a line that is
parallel to the line whose equation
is 2 3 12x y ?
(a) 6 4 2y x (c) 4 6 2x y
(b) 6 4 2y x (d) 6 4 2x y
4) Which equation represents a line that is
parallel to the line y x 3 2 ?
(a) 4 2 5x y (c) y x 3 4
(b) 2 4 1x y (d) y x 4 2
5) Find the equation of the line parallel to the line
whose equation is 2y – 4x = 10 and which passes
through the point (1, 2).
6) Find the equation of the line perpendicular to
the line whose equation is y = 5
6x – 4 and
which passes through the point (5, 3).
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7) Write an equation of a line that is parallel to
5 15y x and passes through (1, 8).
8) Write an equation of a line that is
perpendicular to 2
65
y x and passes
through (10, –17).
9) Write an equation of a line that is parallel to
2 7y x whose y – intercept is –3.
10) Write an equation of a line that is
perpendicular to 3 5y x whose y –
intercept is –3.
11) Write an equation of a line that is parallel to:
29
3y x
.
12) Write an equation of a line that is
perpendicular to the line below.
510
6y x