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Category 3
STAAR Review – 2017
•Day 6
Essential Vocabulary
• Independent
• Dependent
• Variable
• Equation
• Table
• Inequality
• Mapping
• Graph
y = 3x + 5
y = 3x + 5
y = 3x + 5
y = 3x + 5
y < 3x + 5
X
Y
-1
2
0
5
1
8
2
11
-1
0
1
2
2
5
8
11
Domain and Range
• Domain is the set of all x values
• Range is the set of all y values
• To find domain: examine the right and left boundaries
of the function
• To find range: examine the top and bottom boundaries
of the function
• Whenever a function has two boundaries, both signs
should be less than (< or ≤).
2, Ab2B
Definition
a. Specific points
b. Linear
c. Quadratic
d. Exponential
Example
a. {(3, 6), (2, 8), (5, 3)}
b. y = 3x + 2
c. y = x2 + 5x + 4
d. y = 2(3)x
Domain
All the possible
x-coordinates.
a. { 3, 2, 5}
b. -∞ < x < ∞ (all real numbers)
c. -∞ < x < ∞
d. -∞ < x < ∞
Range
All the possible
y-coordinates.
a. { 6, 8, 3}
b. -∞ < y < ∞
c. y ≥ 4
d. y ≥ 0
Domain and Range
Graphs
• The domain of a function is the set of all the xcoordinates in the functions’ graph
Domain
3 ≤ x ≤ 12
The range of a function is the set of all the y-coordinates in the
functions’ graph
Range 6 ≤ y ≤ 12
Domain and Range of a Graph
What is the domain of this function?
What is the range of this function?
Domain is 0 ≤ x ≤ 4
Range is 1 ≤ y ≤ 5
The average daily high temperature for the month of May is
represented by the function t = 0.2n + 80 Where n is the date of the
month. (May has 31 days.)
n
t
Min. days in month
0
80
Min. temp.
Max. days in month
31
86.2
Max. temp.
What is a reasonable estimate of the domain?
Answer: 0 ≤ n ≤ 31
What is a reasonable estimate of the range
Answer: 80 ≤ t ≤ 87
Put equation in
calculator and
knowing minimum
and maximum
days in a month
(n), find the min &
max for t.
Discrete & Continuous Data
• Discrete Function has a graph with isolated points.
•
Domain: x = 1, 2, 3, 4, 5
Cost of renting DVD’s
-Can’t have parts of
DVD’s.
• Continuous Function has a graph that is unbroken.
•
Domain: x > 0
Face length & ear
length is measured –
can have fractions.
Parallel & Perpendicular Lines
• Parallel Lines have equal slope (m)
y = ¼ x – 3 and
𝟏
𝟒
y=¼x+6
m = for both
• Perpendicular Lines have opposite reciprocal slope (m)
y = ¼ x – 5 and y = -4x + 15
m=
𝟏
𝟒
& m = -4
• Lines with the same y intercept and same slope are the same line.
y = ¾ x – 9 and y = ¾ x – 9
m = ¾ b = -9
3, Ac2C
Parallel and Perpendicular Lines
6
5
4
3
2
1
• Parallel Lines
• have the same slope (m)
-6 -5 -4 -3 -2 -1
Rise
m
Run
• Perpendicular Lines
• have opposite reciprocal slopes
1

2
2

1
-6 -5 -4 -3 -2 -1
7, G.07B
y
1 2 3 4 5 6
-1
-2
-3
-4
-5
-6
6
5
4
3
2
1
-1
-2
-3
-4
-5
-6
1

2
1

2
x
1

2
y
x
1 2 3 4 5 6
2

1
What is the equation of the line parallel to y = 3x + 2 and passes through the point (1,8).
Parallel lines have same slope
Slope of new line is 3
Use point slope form (y –y1) = m(x – x1)
(y – 8) = 3(x – 1)
y – 8 = 3x – 3
ANSWER
y = 3x + 5
What is the equation of the line perpendicular to y = -4x + 2 and passes through the point (2,-3)
Perpendicular lines have opposite reciprocal slopes (opposite & flip)
Slope of new line is
1
4
Use point slope form (y –y1) = m(x – x1)
1
(y + 3) = (x – 2)
y+3
ANSWER
4
1
= x
4
1
4
–
1
2
y= x-3
1
2
• What is the slope of a line parallel to the x axis?
• Write the equation of the line.
M=0
Y=3
• What is the slope of a line perpendicular to the x axis?
• Write the equation of the line.
• What is the slope of a line parallel to the y axis?
• Write the equation of the line.
M = undefined
X=5
M = undefined
X=7
• What is the slope of a line perpendicular to the y axis?
• Write the equation of the line.
M=0
Y=4
•Day 7
Direct Variation
• 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
Direct Variation
• Direct Variation ALWAYS goes through the origin, the point (0,0).
• Direct variation works the same as a proportions.
Important Vocabulary
• Equation
Must have an = sign
• Inequality
Must have <, >, ≤, ≥
• Substitution
To replace a variable with a number
• Inverse Operations
• System of Equations
• Solution
• Perimeter
Method of solving an equation
More than 1 equation
Answer to an equation
The distance around an object
Perimeter Rectangle = 2l + 2w
4, Ac4B
Various Forms of a Linear Equations
y  mx  b
Slope-Intercept Form
m  slope of the line
b  y  intercept
Ax  By  C
Standard Form
A, B, and C are integers
A  0, A must be postive
y  y1  m  x  x1 
Point-Slope Form
m  slope of the line
 x1 , y1  is any point
Writing Equations and Inequalities
• Identify if the situation warrants an
equation (=) or an inequality (<, >, ≤, ≥).
• Equations are used when quantities are
equal.
• Inequalities are used when quantities are
not equal.
Look for words like:
4, Ac3A
No more than
No less than
At least
At most
Writing Equations from Graph
m = slope of line
b = y-intercept
Y = mx + b
2
y  x4
3
Rise 2
Run 3


Write equations Using Regressions
• 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
function type.
and arrow over to CALC and choose the appropriate
Press Enter to view equation.
1, Ab1B
Writing Equations from Tables
• 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
25 150
Answer: c = 5b + 25
3, Ac1C
Linear or Quadratic or Exponential
• How do I know what type of function to use?
• STAAR questions will be either
• Linear: LinReg, ax+b
• Quadratic: QuadReg, ax2 + bc + c
• Exponential: ExpReg, a(b)x
• If you aren’t sure, look at the answers and see if they are
linear (y = x) or quadratic (y = x²) or exponential (y = abx)
Writing 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
3
-3
1
Answer: y  x  2
3
y
-1
-3
3, Ac2D
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 = 12
-3x
-3x
2y = -3x + 12
2
2
2
3
y   x6
2
3, A.05C
Step 1: Move the x
Step 2: Divide
everything by the
number in front of y
Writing Equations Given Slope and a Point
Given a point (-3,7) and m = -2, write the equation of
the line.
y  y1  m(x  x1)
y  7  2(x  (3))
y  7  2(x  3)



What if the
directions said to
write the equation
in slope-intercept
form?

Point Slope Form
This is the equation of the
line in point-slope form!!
y  7  2x  6
+7
+7
y  2x 1
This is the equation of the
line in slope intercept form!!
Write Equations from Situation
• Example: Identify the situation that best represents
the amount f(n) = 425 + 50n.
Slope (rate of change) =
Y intercept (initial value) =
50
425
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.
• Example: Carmen receives a $50 gift card to the local
movie theater. Each movie she watches costs $6.75.
Which table best describes b, the balance on her gift card
after she watches m movies?
A
1. Write an equation b = 50 – 6.75m
2. Put equation in y=
C
1, Ab1D
m
b
m
b
0
0
0
50
1
6.75
1
56.75
5
33.75
5
83.75
7
47.25
7
97.25
m
b
m
b
1
43.25
0
50
2
36.50
1
43.25
4
29.75
5
16.25
7
23
7
2.75
B
D
Solving Equations
• TO SOLVE EVERY EQUATION: (with one variable)
• Ask yourself this EVERY time you solve an equation. Should I…..
•
•
•
•
•
•
•
1. Do Distributive Property.
2. Combine Like Terms.
3. Move all variables to one side.
4. Add/subtract on both sides.
5. Multiply/divide on both sides last.
***Goal – to get x by itself
6. Check the answer for reasonableness and accuracy.
Solving Equations
• Example: Mr. Jones charges $25 for an estimate plus $18
per hour to repair dents in car doors. The total charge for a
repair is $151. How many hours did Mr. Jones work on the
repair?
151 = 18h + 25
A. 5 hours
B. 6 hours
C. 7 hours
D. 8 hours
4, Ac3B
1. Write the equation
2. Substitute each answer
for “h”
Solving Inequalities
• Convert inequalities from Standard form (Ax + By > C)
to slope-intercept form (y = mx + b).
• Use the same steps as you would for an equation, but
remember that if you multiply or divide by a negative
number, you must flip the inequality sign!
• Example: 4x – 2y ≤ 5
- 4x
- 4x
-2y ≤ -4x + 5
Because you
-2 -2 -2
divided by a
negative, you
must flip the ≤
to !
y  2x –
𝟓
𝟐
•Day 8
Writing Systems of Equations
• Most systems are comprised of 2 types of equations
• A total equation that represents the total number of items
• And a comparison equation that represents the relationship
between the two variables
• Identify what the variables represent
• Identify which numbers go with each equation type.
Writing Systems of Equations
• Example: Claudia purchased 12 shirts and jeans for
the school year. Jeans cost $22 and shirts cost $15.
If Claudia spent a total of $215, write a system of
equations that could be used to find the number of
shirts that Claudia purchased.
j = jeans
s = shirts
Total Equation
Comparison Equation (money)
j + s = 12
22j + 15s = 215
Writing Systems of Equations
• When rectangles enter the picture:
• Remember: Perimeter = 2l + 2w
• Example: The length of a rectangle is three times the width.
The perimeter of the rectangle is 16 inches. Which system
of equations can be used to determine the dimensions of
the rectangle?
A. l = 3 + w
2l + 2w = 16
C. l = 3w
l + w = 16
B. l = 3w
2l + 2w = 16
D. l = w – 3
l + w = 16
Write Linear System Given a Table
• Use STAT to enter the first two columns from the given
table.
• Find the equation for y1
• Use STAT to enter the first and last columns from the
given table.
• Find the equation for y2
• These two equations are the system.
• Notice the solution to the system is in the table – the
point (6,7)
Write Linear System given a Graph
m=3
b = -2
y = mx + b
y = 3x - 2
The system is:
Y = 3x – 2
Y = - 3/4x + 3
m = -3/4
b=3
y = mx + b
y = - 3/4 x + 3
Solving Systems of Equations
• The solution to a system of linear equations is the point
where the two lines intersect.
• If you are unsure how to solve a problem by
substitution, elimination, or graphing, you can
substitute each answer choice into the equations to
see which one works for BOTH equations.
Solving Systems by Graphs
Solving a System with the graphing
calculator
•
•
•
•
•
•
•
•
•
•
Put equations in Y = mx + b form.
Enter first equations into Y1=
Enter second equation into Y2=
Press Graph (make sure you can see the
intersection – if not, press Zoom – Zoom
Out Enter, Enter)
Press CALC (2nd – Trace)
#5 : Intersect
Enter
Enter
Enter
Intersection: X= and Y=
To solve a system graphically – equations
must be in slope intercept form (y = mx + b)
Solve the system:
y=x+1
Y = 2x + 4
Solution:
(-3, -2)
Solving a Linear System by Substitution
Solving Systems by Elimination
Solving Systems by Elimination
(2, 2)
When to use which method……
Solve the system
Write Linear Inequality Given a Graph