© a very good teacher 2007 algebra 1 eoc preparation unit objective 2 algebra 1 eoc preparation...
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© A Very Good Teacher 2007
Algebra 1
EOC Preparation UnitObjective 2
Algebra 1
EOC Preparation UnitObjective 2
Student Copy
Independent and Dependent Quantities
Independent and Dependent Quantities must be variables (letters), not constants
(numbers).
Independent Quantities are often quantities that cannot be controlled
Dependent Quantities change as a result of the Independent Quantities
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Algebra 1
EOC Preparation UnitObjective 3:
© A Very Good Teacher 2007
Interpreting Linear Functions Functions can be represented in different ways:
y = 2x + 3 means the same thing as f(x) = 2x + 3
Linear Functions must have a slope (rate of change) and a y intercept (initial value).
In a function… the slope is the constant (number) next to the
variable the y intercept is the constant (number) by itself
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© A Very Good Teacher 2007
Interpreting Linear Functions, cont…
• Example: Identify the situation that best represents the amount f(n) = 425 + 50n.
Slope (rate of change) =
Y intercept (initial value) =
50425
Find an answer that has:
425 as a non-changing value and
50 as a recurring charge every month, every year, etc…
Something like Joe has $425 in his savings account and he adds $50 every month.
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© A Very Good Teacher 2007
Functions and their equations
• To find the equation of a function when you are given the table, use the feature of your graphing calculator.
Enter the table into the calculator using L1 for x and L2 for y.
Then return to and arrow over to CALC and choose the appropriate function type.
Press Enter to view equation.
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© A Very Good Teacher 2007
Functions and their equations
• How do I know what type of function to use?
• All EOC questions will either be Linear (LinReg, ax+b) or Quadratic (QuadReg)
• If you aren’t sure look at the answers and see if they are linear or quadratic.
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© A Very Good Teacher 2007
Functions and their equations
• Here’s one to try:The table below shows the relationship between x and y. Which function best represents the relationship between the quantities in the table?
x y
-1 -1
0 -4
1 -1
2 8
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A. y = 2x² - 4
B. y = 3x² - 4
C. y = 2x² + 4
D. y = 3x² + 4
© A Very Good Teacher 2007
Converting Tables to Equations• When given a table of values, USE STAT!• Example: What equation describes the relationship
between the total cost, c, and the number of books, b?
b c
10 75
15 100
20 125
25 150Answer: c = 5x + 25
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© A Very Good Teacher 2007
Converting Graphs to Equations• Make a table of values
• Then, use STAT!
• Example: Which linear function describes the graph shown below?
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x y
Answer: y = -.5x + 4
-2 5
0 4
2 3
4 2
© A Very Good Teacher 2007
Converting Equation to Graph
• Graph the function in y =
• Example: Which graph best describes the function y = -3.25x + 4?
Find an answer that has the same y intercept and x intercept as the calculator graph.
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© A Very Good Teacher 2007
Equations that are in Standard Form• Sometimes your equations won’t be in
y = mx + b form.
• They will be in standard form: Ax + By = C• You must convert them to use the calculator!
Example: 3x + 2y = 12Step 1: Move the x -3x -3x
2y = -3x + 12 Step 2: Divide everything by the number in front of y
2 2 2
36
2y x
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© A Very Good Teacher 2007
Slope and Rate of Change (m)• Slope and rate of change are the same thing!
• They both indicate the steepness of a line.
• Three ways to find the slope of a line:
By Formula: By Counting: By Looking:
2 1
2 1
y ym
x x
rise
mrun
y x bm
You must have 2 points
on a line
You must have a graph
You must have an equation
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© A Very Good Teacher 2007
Slope and Rate of Change (m), cont…• By Formula:
• Find two points on the graph (they won’t be given to you)
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2 1
2 1
y ym
x x
(0, 4) and (2, 3)1x 2x1y 2y
1
2
3 4
2 0
© A Very Good Teacher 2007
Slope and Rate of Change (m), cont…
• By Counting
• Find two points on the graph
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risem
run
Down 2
Right 4
2
4
1
2
© A Very Good Teacher 2007
Slope and Rate of Change (m), cont…• By Looking
• The equation won’t be in y = mx + b form
• You’ll have to change it• If in Standard Form use Process on Slide 7• If in some other form, you’ll have to work it out…
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Example: What is the rate of change of the function
4y = -2(x – 24)? Try to get rid of any parentheses and get the y by itself (isolated).
4y = -2x + 484 4 4
112
2y x
1
2m
© A Very Good Teacher 2007
Slope and Rate of Change (m), cont…
• Special Cases
• Horizontal lines line y = 4
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• Vertical lines like x = 4
Have slope of zero, m = 0
Have slope that is undefined
© A Very Good Teacher 2007
m and b in a Linear Function• Changes to m, the
slope, of a line effect its steepness
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• Changes to b, the y intercept, of a line effect its vertical position (up or down)
y = 1x + 0
y = 3x + 0
y = 1/3 x + 0
y = 1x + 0
y = 1x + 3
y = 1x - 4
© A Very Good Teacher 2007
m and b in a Linear Function, cont…
• Parallel Lines have equal slope (m)
y = ¼ x – 3 and y = ¼ x + 6
• Perpendicular Lines have opposite reciprocal slope (m)
y = ¼ x – 5 and y = -4x + 15
• Lines with the same y intercept will have the same number for b
y = ¾ x – 9 and y = 5x – 9
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© A Very Good Teacher 2007
Linear Equations from Points• Make a table
• USE STAT
• Example: Which equation represents the line that passes through the points (3, -1) and (-3, -3)?
x y
3 -1
-3 -3
Answer: 1
23
y x
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© A Very Good Teacher 2007
Intercepts of Lines• To find the
intercepts from a graph… just look!
• The x intercept is where a line crosses the x axis
• The y intercept is where a line crosses the y axis
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(4, 0)
(0, 2)
© A Very Good Teacher 2007
Intercepts of Lines, cont…• To find intercepts from equations, use your
calculator to graph them
• Example: Find the x and y intercepts of 4x – 3y = 12.
-4x -4x
-3y = -4x + 12-3 -3 -3
44
3y x
x intercept: (3, 0)
y intercept: (0, -4)
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© A Very Good Teacher 2007
Direct Variation• Set up a proportion!
• Make sure that similar numbers appear in the same location in the proportion
• Example: If y varies directly with x and y is 16 when x is 5 what is the value of x when y = 8?
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16x = 5(8)
16x = 4016 16
x = 2.5
y y
x x 16 8
5 x
© A Very Good Teacher 2007
Direct Variation, cont…• To find the constant of variation use a
linear function (y = kx) and find the slope
• The slope, m, is the same thing as k• Example: If y varies directly with x and y = 6
when x = 2, what is the constant of variation?
y = kx
6 = k(2)2 2
3 = k
The equation for this situation would be y = 3x
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