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Linear
Functions
Slope
Parallel &
Perpendicular
Lines
Different
Forms of
Linear Equations
$100 $100 $100 $100
$200
$200 $200 $200
$300 $300 $300
Solving Equations
and Inequalities
Review
Misc. Linear
Functions
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$200
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$300
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$400 $400 $400 $400
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$500 $500 $500 $500
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$100 Question Linear Functions
Explain how you know if a graph
represents the following:
Function:
Linear function:
Direct Variation (Proportional Relationship):
$100 Answer Linear Functions
Function: passes VLT
Linear function: passes VLT & straight line
Direct Variation (proportional relationship:
passes VLT, straight line & crosses through
(0,0)
$200 Question Linear Functions
Which table(s) are linear?
Explain how you know.
A.
B.
$200 Answer Linear Functions
B.) x and y are both going up
with constant rates
* same rate of change *
Rate of change = change of y
change of x
Table A – not a constant rate of change
$300 Question Linear Functions
Which of the following equations is not linear?
Explain or show how you know.
A) y = 2x2 – 7
B) -6 = y
C) 4x – 2y = 10
D) y = 3x + 1
$300 Answer Linear Functions
A) y = 2x2 – 7 not linear--exponent
B) -6 = y
horizontal—straight line
C) 4x – 2y = 10 standard form x and y-int. --line
D) y = 3x + 1 slope-int.—always form a line
$400 Question Linear Functions
Explain if each equation represents a direct
variation (proportional relationship).
Identify the constant of variation
if it is a direct variation.
A.) 3y = 4x + 1
B.) y + 3x = 0
$400 Answer Linear Functions
Solve each equation for y. Look to see if it’s in
y = kx form
k = constant of variation
A.) 3y = 4x + 1
y = 4/3x + 1
Not in y = kx form (+1)
B.) y + 3x = 0
y = -3x
Yes, in y = kx form
k = m -3
$500 Question Linear Functions
Let y = 2x + 9. If the value of x increases
by 6, which of the following best describes
the change in the value of y.
a.) decreases by 6
b.) increases by 6
c.) increases by 12
d.) increases by 21
$500 Answer Linear Functions
As x goes up by 6
The value of y. c.) increases by 12
$100 Question Slope
Find the slope of the line
(0, 4)
(3, -5)
$100 Answer Slope
Rise
Run
m = -3
$200 Question Slope
Write the equation of the line that passes
through each pair of points in slopeintercept form
(4,6) and (2, 2)
$200 Answer Slope
m=2
Remember a neg.
divided by a neg. is
positive
1. Label Points and use slope formula
2. Choose a point and use point-slope formula
y – 6 = 2(x – 4) Distribute
y – 6 = 2x – 8 Add 4 to both sides
y = 2x - 2
$300 Question Slope
Put the following equation into slopeintercept form. Identify the slope and yintercept. Then use the slope and y-int. to
graph the line.
3x – y = 2
$300 Answer Slope
3x – y = 2
-3x
-3x
-y = -3x + 2
-1 -1 -1
y = 3x -2 m = 3
b = -2
$400 Question Slope
Laurel graphed the equation
y = -2x + 5. Katelyn then graphed an
equation that was a line that was not as
steep as Laurel’s. Which equation could
have been the one Katelyn graphed?
a.) y = -3x + 5
b.) y = 1/2x + 6
c.) y = 4x – 2
d.) y = -2x + 3
$400 Answer Slope
B.) y = 1/2x + 6 is not as steep.
Fractions (between -1 and 1: non-improper)
are less steep than any integer—
even if it’s negative.
$500 Question Slope
The cost of hiring Zach as a painter is
given by the linear equation
C = 10h + 100,
where h is the number of hours Zach
works. Identify the slope and y-int.
What does the slope of the line represent?
What does the y-intercept represent?
$500 Answer Slope
m = 10 The slope means Zach earns $10
per hour.
b = 100 The y-intercept represents base
charge of hiring Zach
$100 Question Parallel &
Perpendicular Lines
What are two different ways that lines can
be perpendicular?
$100 Answer Parallel &
Perpendicular Lines
Vertical lines are perpendicular to a
horizontal lines. Ex. x = 3 and y = -2
When the product of slopes = -1
Ex. 4 and -1/4
$200 Question Parallel &
Perpendicular Lines
A line has the equation x + 2y = 5
What is the slope of a line parallel to this
line?
a.) – 2 b.) - ½ c.) ½
d.) 2
$200 Answer Parallel &
Perpendicular Lines
A line has the equation x + 2y = 5
1. Put line in slope-int. form y = -1x + 5
2
2
2. Parallel -- same slope -- b.) - ½
$300 Question Parallel &
Perpendicular Lines
$300 Answer Parallel &
Perpendicular Lines
A. 1 and -1 are “opposite reciprocals”
$400 Question Parallel &
Perpendicular Lines
$400 Answer Parallel &
Perpendicular Lines
Line AB has a slope of 1 and Line BC
has a slope of -3/2 and Line AC has a
slope of 0. None of the slopes will
have a product of -1 so D is the answer
$500 Question Parallel &
Perpendicular Lines
Write an equation that is perpendicular
to the given line below that passes
through the point (- 6, 2)
$500 Answer Parallel &
Perpendicular Lines
1. Slope will be -3 (opp. reciprocal)
2. Use point-slope form
y- 2 = -3(x – (-6)) Distributive Prop.
y = -3x -16
$100 Question Different Forms
of Linear Equations
Find and use the x and y intercepts to
graph the line.
-x + 3y = 6
$100 Answer Different Forms of
Linear Equations
-x + 3y = 6
-x + 3y = 6
0 + 3y = 6
-x + 3(0) = 6
y-int. = 2
-x = 6
(0,2)
x-int. = -6
(-6,0)
(0,2)
(-6,0)
$200 Question Different Forms
of Linear Equations
Find and use the x and y intercepts to
graph the line.
-2x = 12 + 4y
$200 Answer Different Forms of Linear Equations
-2x = 12 + 4y
-4y from both sides
-2x - 4y = 12
Now in standard form
x-int. = (-6,0)
y-int. = (0,-3)
(-6,0)
(0, -3)
$300 Question Different Forms
of Linear Equations
What is the x-intercept of the linear
function f(x) = -3x + 6?
Note: f(x) is dependent variable (y)
a.) -2
b.) 2
c.) 3
d.) 6
$300 Answer Different Forms of
Linear Equations
f(x) = -3x + 6 Think: y = -3x + 6
add 3x to both sides –standard form
y + 3x = 6
0 + 3x = 6
x-int. = 2 (b)
$400 Question Different Forms
of Linear Equations
A line has a slope of
and passes
through the point (-3, 4).
What is the equation of the line in
point-slope form?
What is the equation of the line in
slope-intercept form?
$400 Answer Different Forms of
Linear Equations
Point-slope form
y- 4 =
[x – (-3)]
y–4=
(x + 3)
Slope-Intercept Form y =
x+6
$500 Question Different Forms
of Linear Equations
Is every linear relationship a direct
variation? Is every direct
variation/proportional relationship a linear
relationship? Explain.
$500 Answer Different Forms of
Linear Equations
Every linear relationship is not a direct
variation—only if the y-int. is 0. However,
every direct variation is linear because it has a
constant rate of change.
Direct Variation/prop. relat.: y = 3x (also linear)
Not a direct variation y = 3x +5 (is linear)
$100 Question Solving Equations and
Inequalities Review
Solve and Graph
-2m < 8
$100 Answer Solving Equations and
Inequalities Review
-2m < 8
- 2 -2
m > -4
Open to the right
$200 Question Solving Equations and
Inequalities Review
Solve and Graph
4a – 10 > 6a
$200 Answer Solving Equations and
Inequalities Review
4a – 10 > 6a
-10 > 2a
-5 > a
Open to the left
4 y
8y
2
2
7
7
$300 Question Solving Equations and
Inequalities Review
Solve and Graph
$300 Answer Solving Equations and
Inequalities Review
Closed to the right
$400 Question Solving Equations and
Inequalities Review
Solve.
2(t - 4) = 3 – (3t + 1)
$400 Answer Solving Equations and
Inequalities Review
2(t - 4) = 3 – (3t + 1)
2t – 8 = 3 – 3t – 1
2t – 8 = 2 – 3t
5t = 10
t=2
$500 Question Solving Equations and
Inequalities Review
Janine has job offers at two companies. One
company offers a starting salary of $28,000
with a raise of $3000 each year. The other
company offers a starting salary of $36,000
with a raise of $2000 each year.
Find when the two offers are the same.
Which offer will be better if you plan to stay
longer than 10 years?
$500 Answer Solving Equations and
Inequalities Review
$28,000 + $3000x = $36,000 + $2000x
Solve for x. x = 8 years. They will both make
$52,000
After 8 years, the starting rate of $28,000 with
a increase of $3000 per year will earn more $.
$100 Question Miscellaneous
The table shows an employee’s pay per hour.
Determine if there is a direct variation between
the pay and number of hours worked. If so, find
the equation of direct variation
$100 Answer Miscellaneous
You can use the ratio
to check.
17/2 = 8.5 and 34/4 = 8.5 ratios are =
(proportional) so it’s a direct variation
The equation would be y = 8.5x
$200 Question Miscellaneous
Make a table and graph for y = -x + 3
Is this equation linear? Explain.
$200 Answer Miscellaneous
Graph has a negative slope—straight line
It is a linear function
because table has a
constant rate of change
and graph is a straight
line
$300 Question Miscellaneous
Write the equation of the line that passes
through each pair of points in slopeintercept form
(-1, 5) and (2, -4)
$300 Answer Miscellaneous
1.) Find Slope m = -3
2. ) Choose a point (-1,5)
Insert into slope-int. and
solve for b OR
Use point-slope form then
solve for y
Y = -3x + 2
$400 Question Miscellaneous
Also, explain what each intercept represents
$400 Answer Miscellaneous
x-intercept (8,0)
Namia has enough
money to buy 8 lg.
popcorns and 0 drinks
4-intercept (0,20)
Namia has enough
money to buy 20 small
drinks and 0 popcorns
$500 Question Miscellaneous
Prove that ABCD is a parallelogram
$500 Answer Miscellaneous
ABCD is a parallelogram because opposite sides
are parallel.
Line AB and Line DC both have a slope of -2/3
and Line AC and Line DC both have an
undefined slope (vertical lines)
Final Jeopardy
Write an equation in slope-intercept
form for the given table. Explain what
the slope and y-intercept form represent.
Then find the cost for 60 additional
minutes.
Find the rate of change for slope
m = 2/5 or 0.4 (0.40 for each add. min.)
Pick a point and use slope formula or
solve for b to get slope intercept form
y = 0.4x + 40
y –int. means it cost $40 with 0 add. min.
y = 0.4(60) + 40 It will cost $64 for 60
additional minutes.