Summary of 2.1

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Summary of 2.1

• Ordered pair (x, y) (abcissa, ordinate)
• (rt/left, up/down)
• Cordinate plane (0,0) origin
• Linear equation
• slope intercept form y mx b
• point-slope form y y1 m (x x1)
• standard form Ax By C
• To find x-intercept, let y 0 and solve for x.
  Why?
• To find y-intercept, let x 0 and solve for y.
  Why?
• Vertical lines
• x , m undefined, may not have
  y-intercept
• Horizontal lines
• y , m 0, may not have an x-intercept

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Ex. 1 2x 3y 6Ex. 2
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2.2 Slope of a LineObjective To determine
the slope of a line and understand parallel and
perpendicular lines.
Slope
Positive slope -
Rises from left to right
Falls from left to right
Negative slope -
Zero slope -
Horizontal line
Undefined slope -
Vertical line
Slopes of perpendicular lines have negative
reciprocal slopes. What kind of slopes do
parallel lines have? Ex. 1
4
b. Points (2, 7) and (-3, 4)

• From an equation y 4 3x or 2x
  5y 7

• Ex. 3 Solve for the missing variable.
• (a, 3) (2, 4) and m -2 Find a.
• (4, a) (6, 3a) and m 2/3 Find a.
• (3, 7) (x, x2 - 2) and m -4 Find x.

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Can you determine the slope of a line and
determine the slope of parallel and
perpendicular lines? Assign 2.2 1-6, 15-33
odd, 36-39, 44, 45, 47-52, 59, 71, 72
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Warm-Up 2.3

• Write an equation for the horizontal line through
  (2, 7).
• Write an equation for the vertical line through
  (-5, 6).
• Find the x and y-intercept for 2x 3y 9 .

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2.2 Slope of a Line

• Slope
• Slope 0
• Slope
• 36. Slope
• Slope -2
• A m 0 B m 2 C m -1 D m
• 20 minutes before the water boils.
• Slope of C is 100 C per minute.
• There is no temperature change.
• 72. 1

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2.3 The Equation of a LineObjective To write
equations for lines using slope-intercept,
standard and point-slope form.
Slope Intercept y mx b
Ex1) m 2 b 3 Ex2) (2, 1) m -2 Ex3)
(-3, 1) (4, 2)
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Point Slope formula y y1 m(x
x1) Ex4) m ½ (2, -3) Ex5) (3, -2) (1, 4)
Standard form Ax By C (Remember A, B
and C must be integers.)
Ex7) Write an equation in standard form that is
parallel to 3x y 7 and passes
through (2, -4).
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Ex8) Write an equation perpendicular to 3x 2y
4.
Ex9) Write an equation for the perpendicular
bisector of the segment with endpoints
(2, -3) (4 , 9).
Can you use the given information to write
equations for lines using slope-intercept,
standard and point-slope form?Assign 2.3
3-17 odd, 24-36 (x3), 43-51 odd, 55, 65, 67
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WARM-UP
The equation for the height of a ball thrown into
the air at any time is given by h -16t2 64t
24. Where t is the time in seconds and h is the
height of the ball. 1. When will the ball hit
the ground? 2. What values of time are valid
for this situation?
Fix s for next yearthis one doesnt factor!!
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2.3 The Equation of a Line

• 24. y 2x - 3
• 30. y
• 36. 4x 3y -30

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2.4 Linear InequalitiesObjective To graph
linear and absolute value inequalities.
A linear inequality separates the coordinate
plane into 2 half-planes separated by a boundary
line.
If the boundary line is included (lt gt) it is a
_______ line.
solid
If the boundary line is not included ( lt gt) it is
a ________ line.
dotted

• Hints
• Graph the lines as an equation.
• Boundary line solid/dotted
• Shade appropriate side of boundary line

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Can you graph linear and absolute value
inequalities? Assign 2.4 1-29 odd, 53-57 odd
2.1 67-70
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Warm-UP

• Correct 3 problems
• that you missed on your
• Chapter 1 Test

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2.4 Solutions

• 68. y x2 3
• 70. y

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Today is the quiz2.1-2.4

• Staple your homework in the following order.
  Put it in the tray.
• 2.1
• 2.2
• 2.3
• 2.4
• Review

50 points
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2.5 Introduction to Functions
Objective To understand what it takes to be
considered a function and to give the domain and
range for relations and functions.

• A relation is any set of ordered pairs. It can
  be represented in various ways.

? Ordered pairs (2, 1), (0, 2), (1, 2), (0,
5)

• Table of values

? Mapping diagram
2
1
1
2
0
5
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A function is a relation that pairs each x-value
with exactly one y-value. (ALL X-VALUES MUST BE
DIFFERENT.)
Go back and give the domain and range for the
relations on the previous slide.
Input x-values domain Output y-value range
Function?
Ex. 1 (2, 1), (-4, 0), (-3, 1), (3, -1)
YES Why?
Ex. 2 (4, -1), (5, 3), (4, -2), (7, 1)
NO Why?
A graph that passes the vertical line test (VLT)
is a function. A vertical line can cross the
graph at most 1 time. Meaning, each x-value has
a unique y-value. Give the domain and range for
each.
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Do you understand what it takes to be considered
a function and how to give the domain and range
for relations and functions?
Homework 2.5
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Warm Up 2.6
Determine whether each is a function. Explain
why or why not? Give the domain and range of
each.
1) y x 2 2) (2, 3), (0, 1), (4, 2),
(0, -2)
3) 4) 5)
How can you easily tell if an equation represents
a function? y x2 3 x2 y2 4 x y2 2y
1 y sin x
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2.5 - Solutions

• Yes 56. Domain all real s
• 12. Yes Range y y gt-3
• 14. Yes
• 16. No
• 18. No
• 22. Domain x -2 lt x lt 2
• Range y -3 lt y lt 3
• Domain x -2 lt x lt 2
• Range y 2 lt y lt 6

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2.6 Function NotationObjective To evaluate
equations in function notation.

• An equation such as y 3x 1 can be written in
  function notation.

Replace y with f(x). When we read this we say
f of x . So. y 3x 1 is the same as
f(x) 3x 1.
Ex. 1 Given f(x) 3x 1 f(0) f(a)
f(-2) f(m2) f(3x) f(a 1)
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• f(3x) b) g(1) h(8)
• c. f(-3) 2h(10) d) f(x) 2g(x2)

Now lets do some composition of functions.
Composition of functions can be written in one of
2 ways(This means to substitute one function
into another function.) f(g(x)) or f o g(x)
Ex. 3 f(g(x)) g(f(x)) f o h(x)
h(f(x))
Is composition of functions commutative? Explain.
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Ex. 5 Given f(x) 3x2 1 a. 2f(x) 5 b.
3f(x-1) 2
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f(x) y (x, y) f(1) 3 (1, 3)
Ex. 7 Use the graph to answer the following. f(3)
f(0) f(-2) f(x) 2 f(x) 0 f(x) 4
Can you evaluate equations in function
notation?Assign 2.6 3-57 (x3), 71, 72,
77-83 odd
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2.6 Function Notation

• -13 12. a2 15a 58
• 18. -3 24. 2a 3
• f(p) 0 36. -4x2 -2x 4
• 42. -1 48.
• 54. 15
• 72. a. 1
• b. 3
• c. 0
• d. 2

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E Quiz over 2.6
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2.7 Operations on Functions
Objective To continue using function notation
and to learn how to perform operations on
functions
Remember There will be a mini quiz over
FUNctions tomorrow! ?
Given f(x) x2 3, g(x) 2x 1 and
h(x) 4x
Add Ex1) (f g)(x) or f(x) g(x) Ex2)
g(x) h(x)
Subtract Ex3) (f g)(x) or f(x) g(x)
Ex4) h(x) f(x)
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Multiply Ex5) (fg)(x) or f(x) g(x) Ex6) h(a)
g(a)
Dont forget to list your domain issues! Factor
and cancel if possible.
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f(x) x2 x 6 g(x) x 3 h(x) x2
k(x) Ex15) 3g (x2) 4h(x) Ex16)
2g2 (x) h(x) Ex17) 2k(a) g(a) Ex18)
k2(a) 1
Can you use perform operations on functions????
Are you ready for a short quiz over 2.5 and 2.6
Assign 2.7 Worksheet (2.7)
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WARM-UP

• The general form a parabola looks like
• y a(x-h)2 k
• How do you find the vertex?
• Using what you know, sketch the graph of
• How would you graph x (y-2)2 -4?

Think about the domain and range!
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2.6.2 Worksheet Answers
If it can be written as a y .
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• 2.7 Answers
• 31/3 2. (a3 4)/a 3. 1
• a4 2a2 1 5. a2 10a 25 6. 2x3 4
• x2 3 8. x2 6x 9 9. 4/x2
• 1/(3a 1) 11. 16x6 3 12. 16/x
• 50 14. 27a3 27a2 15. (12 4m2)/m2
• (16 24x 9x2)/x2 17. 3a4 13 18. 0
• 19. ½ 20. (3x 4)/x2 21. 3m2 2
• 22. -2x 1 23. 45a2 120 a 81 24. (16
  4a2)/a2

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2.8 Direct and Inverse Variation
Objective To understand the difference between
direct and inverse variation and to solve
equations for each.
Direct Variation As the value of one variable
increases, the other value also increases.
Example The more hours you work, the more pay
you receive.
y kx Where k constant of variation.
Ex If y varies directly as the square of x,
and y 6 when x 2, find k.
1st 2nd
If I ask you to write an equation for this
problem (hint-hint)
Find y when x 3
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The length of a shadow is proportional to the
height of an object.
L kh
Ex If the height of a sign is 6 and the length
of the shadow is 2. a. Find k.
b. If the length of the shadow of the school is
24, how tall is the school?
Method 1 equations
Method 2 proportion
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Inverse Variation
As the value of one variable increases the other
decreases.
Heat from a light is inversely proportional to
the distance from the light.
Note You must still use k and inverse variation
is division.
As the distance increases, the heat decreases.
-OR- As the distance decreases, the heat
increases.

• Ex y varies inversely with x. If x 6 when y
  4,
• find k,
• write an equation,
• find x when y 8, and
• Find y when x 2.

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Other Types of Variation
Do you understand the difference between inverse
and direct variation? Can you solve equations
for both?Assign 2.8 1-25 odd, 29, 31, 35, 37,
42
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2.8 Solutions

• 42. ? 2.83 feet from the center

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Chapter Summary
2.1

• x-intercepts
• y-intercepts
• Horizontal lines and slope of horizontal lines
• Vertical lines and slope of vertical lines

2.2

• Slope from points/graph/equation
• Parallel lines
• Perpendicular lines
• Positive, negative, 0, and undefined slope

2.3

• Standard form Ax By C
• Slope intercept form y mx b
• Point-Slope form y y1 m(x x1)

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2.4

• Linear Inequalities
• Type of line? Solid if lt or gt Dotted if lt or
  gt
• Shade? above (right if vertical) gt or gt
  
  below (left if vertical) for lt
  or lt
• Also know y x 2 y 3 - x y gt x
  2 y lt -x 1

2.5

• Relation
• Function
• Domain input x-values
• Range output y-values
• When Determine if the set is a function and give
  the domain and range for a mapping diagram, set,
  equation or graph.
• Know how to do the above for a continuous or
  discrete graph.

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2.6

• Function Notation f(x) y Replace x with the
  value in ( ) and simplify.
• Composition of functions f(g(x)) f o g (x)
  --- This is when you substitute one function into
  another function and simplify.
• Be able to evaluate functions f(2) y means
  find y when x 2 and
• f(x) 1 means find the x values
  that y 1

2.7

• Operations on functions
• f g f g f g f/g
• Dont forget grouping and simplify your answer.

2.8

• Direct variation is multiply y kx
• Direct variation is also a proportion
• Inverse variation is divide y k/x
• Varies jointly as is multiply what follows y
  kxz
• See page 258 for more examples

? ? Study for your Chapter 2 Test ? ?