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HUSD Middle School Algebra I Final Assessment Study Guide
Page 1 of 10 MDC@ACOE (HUSD) 04/28/13
Finding the x and y intercepts 2 4 8y x+ = − Set y = 0 to find x-intercept: Set x = 0 to find y-intercept:
The x-intercept is ( )2, 0− and the y-intercept
is ( )0, 4−
6.0/8.EE.5, F-IF-7a
Finding the slope and y-intercept of the line
2 4 8y x+ = −
First solve for y:
Find the slope by identify the coefficient of x: 2m = −
Find the y-intercept by identifying the constant:
�
0, − 4( )
6.0/8.EE.5, F-IF-7a
Practice: Find the x and y intercepts. 1) 6 2 12y x+ = 2) 3 9 9x y− =
Practice:
Identify the slope & y-intercept 3) 12 3 9x y− = 4) 6 2 12y x+ =
( )
2 4 82 4 4 8 4
2 8 42 8 42 2 2
4 22 4
y xy x x x
y xy x
y xy x
+ = −+ − = − −
= − −− −= +
= − + −= − −
( )2 4 82 0 4 8
4 82
y xxxx
+ = −+ = −
= −= −
( )2 4 82 4 0 8
2 84
y xy
yy
+ = −+ = −
= −= −
HUSD Middle School Algebra I Final Assessment Study Guide
Page 2 of 10 MDC@ACOE (HUSD) 04/28/13
Graphing: 2 4 8y x+ = − Method 1: Find x & y-intercept: From example 1 ( )2, 0− and ( )0, 4− Plot the points and draw the line Method 2: Make a table of values and plot the points Method 3: Use slope & y-intercept
6.0,7.0/A-REI-10, F-IF-7a
rise change in the coordinatesSlope
run change in the coordinatesyx−= =−
Horizontal line: Vertical Line: Find the slope Find the slope of of the line 3y = the line 2x =−
change inchange in3 31 2030
ymx
=
−=− −
=−
=
Slope is 0 Since division by 0 is not defined, this line has no slope or undefined
6.0/8.EE.5
Practice: Graph these equations.
5) 4 8 16y x+ = −
6) 2y = −
Practice:
Find the slope 7) ( ) ( )2,3 , 2, 5−
8) ( ) ( )1, 4 , 2, 2− −
9) 4x =
( )( )
change inchange in3 12 240
ymx
=
− −=− − −
=
HUSD Middle School Algebra I Final Assessment Study Guide
Page 3 of 10 MDC@ACOE (HUSD) 04/28/13
Write the equation of the line with a slope of 2 and passing through the point ( )6, 3− . Use the point slope form of a line, m is the slope and ( ) ( )1 1, 6, 3x y = −
( )( ) ( )
1 1
3 2 63 2 12
3 3 2 12 32 15
y y m x x
y xy x
y xy x
− = −
− − = −+ = −
+ − = − −= −
Given y = ax2 + bx + c for constants a and b. y b= is a horizontal line with a 0m =
x a= is a vertical line with no slope (undefined slope)
6.0, 7.0
Finding parallel and perpendicular slopes
The slope of line l is – 2 which also can be written m = −2 A line parallel to line l has the same slope so m m= = −1 2 A line perpendicular to line l has the slope m2 such that m m⋅ = −2 1:
m mmm
m
⋅ = −− ⋅ = −− ⋅−
= −−
=
2
2
2
2
12 122
1212
7.0,8.0/G-GPE.5
Practice Write the equation of each line. 10) Find the equation of the line passing through the point ( )1,4− and no slope.
11) Find the equation of the line passing through the point ( )1, 5 with a slope of 0.
12) Slope is 1− and passes through the point ( )1,2
13) Passes through the points ( ) ( )2, 3 and 1, 3− −
Practice:
Find the slope that satisfies each condition Line m is given by the equation 6 2 12y x+ =
14) Find the slope of the line parallel to line m
15) Find the slope of the line perpendicular to line m.
HUSD Middle School Algebra I Final Assessment Study Guide
Page 4 of 10 MDC@ACOE (HUSD) 04/28/13
Solving systems of equations: Method 1: Graphing
Solve the system: 4
2 2x yy x− = −= − −
Graph 4x y− = − by finding intercepts Draw the line using ( )( )0,4 4, 0−
Graph 2 2y x= − − using slope & y-intercept Identify the point of intersection
( )2, 2∴ − is the solution to 4
2 2x yy x− = −= − −
9.0/8.EE.8a
Method 2: Substitution
Solve the system:
Solve for one variable Eq. 1
Substitution- Substitute 3 5x− + for y in Eq. 2 Solve for x Substitute 3x = for y
( )3, 4∴ − is the solution to the system
9.0 /8.EE.8b
Practice Solve the systems by graphing
16) 5
2 3 0x yx y− =+ =
17) 4
2 5x yx y+ =−+ =
Practice
Solve by substitution:
18) 2 4 166 16
x yx y− + = −
+ =−
19) 2 14 2 12x yx y
− + = −+ =
3 52 10x yx y+ =⎧
⎨ − =⎩
3 53 3 5 3
5 3 or 3 5
x yx y x x
y x y x
+ =+ − = −
= − = − +
( )3 53 3 59 54
y xyyy
= − += − += − += −
( )2 - 10
2 - -3 5 102 3 5 10
5 5 105 15
3
x yx xx x
xxx
=+ =
+ − =− =
==
HUSD Middle School Algebra I Final Assessment Study Guide
Page 5 of 10 MDC@ACOE (HUSD) 04/28/13
Method3:Solving systems using elimination
Solve:2 13 2 12x yx y− =+ =
1st Rewrite equations so they have opposite terms. Multiply Eq. 1 by 2 (So –2y and 2y are opposite)
( ) ( )2 2 2 13 2 12x yx y− =+ =
2nd Add the equations vertically: 3rd Solve the equation for x. 4th Solve for y by substitution:
9.0/8.EE.8b
Radical Expressions:
Simplify
Example 1 Example 2 Example 3 Example 4
2.0/8.EE.2, N-RN.2
Practice Solve the systems by using Elimination
20) 3 4 462 2 28x yx y+ =
− − = −
21) A vending machine takes only nickels and dimes. At the end of the day there were three times as many nickels as dimes and a total of $25. How many of each coin were there in the machine?
22) 5 7 14 2 22x yx y+ = −− =
Practice
Simplify each expression
23) 13 832p q
24) 24w3x i 6w2x2
25) 810r
26) 11 8 7 18 2 50− + +
4 2 23 2 127 0 14
7 142
x yx yx y
xx
− =+ =+ =
==
( )3 2 12
3 2 2 126 2 122 6
3
x yyyyy
+ =+ =+ =
==
( )2,3∴ is the solution to the system
18m2n3p6
= 2i3i3im2n2n p3( )2= 3⋅m ⋅n ⋅ p3 2n
= 3mnp3 2n
6xy3 i 18x2 y
= 6i18x3y4
= 2i3i2i3i3x2 y4x
= 2 ⋅3xy2 3x
= 6xy2 3x
2 12 27 36
2 4 3 9 3 6
2 2 3 3 3 6
4 3 3 3 6
6 3
− +
= ⋅ − ⋅ +
= ⋅ − +
= − +
= +
8z3z3
= 8z3z3
= 83z2i33
= 2 ⋅2 ⋅2 ⋅3
9z2
= 2 63z
HUSD Middle School Algebra I Final Assessment Study Guide
Page 6 of 10 MDC@ACOE (HUSD) 04/28/13
Solving Radical Equations: Isolate the radical Square both sides
Check your answer(s)
2.0/8.EE.7, N-RN.1
Solving Quadratic Equations
By Factoring
Use the zero-product property Check your answer(s)
By Completing the Square Finding Square Roots
14.0,21.0/A-SSe.3a, A-REI.4b
Practice Solve
27) 1 6 9x x= + −
28) 4 3 4 3z + =
Practice Solve for x. If the equation has no solution, write no solution.
29) 3 23 2x x x+ =−
30) ( )22 49x− =
31) 2 361 0x − =
32) 23 5 2 0x x− + − =
33) 2 18 2y + =
34) 2 3 5 0y y+ − =
( ) ( )2 2
7 2 1 12
2 1 5
2 1 5
2 1 252 24
12
y
y
y
yyy
+ + =
+ =
+ =
+ ===
( )( )
2
2
3 5 23 5 2 03 1 2 03 1 0 or 2 0
1 or 23
x xx xx xx x
x x
+ =+ − =
− + =− = + =
= = −
5 16 05 16
1651654545
55
4 55
2
2
2
2
mm
m
m
m
m
m
− =
=
=
= ±
= ±
= ± ⋅
= ±
( )( )
2
2
2 2 2
2
2
8 12 08 12
8 4 12 4
4 12 16
4 4
4 44 2
4 2 and 4 22 and 6
x xx x
x x
x
x
xx
x xx x
+ + =+ = −
+ + = − +
+ = − +
+ =
+ = ±= − ±
= − + = − −= − = −
Add -12 to both sides
Add 42 to both sides to complete the square Square root both sides
Simplify radicand
HUSD Middle School Algebra I Final Assessment Study Guide
Page 7 of 10 MDC@ACOE (HUSD) 04/28/13
Solve the equation by using the quadratic formula:
20.0/A-REI-4b
Compound Inequalities
Solve and Graph
Example 1
Example 2
14.0/A-REI.3
Practice Solve for x:
35) 26 4 10x x− =
36) 24 12 7x x+ =
37) 2 2 5x x− =
38) 2
3 1211 1x x+ =
− −
Practice
Solve and graph each inequality or equation:
39) 4 3 15z + ≤
40) 4 3 13x− >
41) 2 3 7y− − ≥
42) 3 2 5 7x− < − + ≤
43) 2 5 73x − =
( ) ( ) ( )
2
2
2
3 4 2 03, 4, 2
42
4 4 4 3 22 3
4 16 246
4 406
4 2 106
2 103
x xa b c
b b acxa
x
x
x
x
x
− − == = − = −
− ± −=
− − ± − − ⋅ ⋅ −=
⋅± +=
±=
±=
±=
Standard form Identify a, b, c Substitute Simplify
-‐5 -‐4 -‐3 -‐2 -‐1 0 1 2 3 4 5
2x −1 ≥ 3
− 2x −1( ) ≥ 3 or 2x −1≥ 3
2x −1≤ −3 or 2x ≥ 42x ≤ −2 or x ≥ 2x ≤ −1
Solutionset x | x ≤ −1 or x ≥ 2{ }
-‐5 -‐4 -‐3 -‐2 -‐1 0 1 2 3 4 5
( )1 2
1 2 or 1 21 2 or 1
3
x
x xx xx
+ <
− + < + <+ > − <
> −
{ }Solution Set 3 1x x− < <
HUSD Middle School Algebra I Final Assessment Study Guide
Page 8 of 10 MDC@ACOE (HUSD) 04/28/13
Graphing Quadratics: Graph the function ( ) 22 4 1f x x x= + − 1st Find the vertex Find the x-coordinate of the vertex Find the y-coordinate of the vertex by substituting the x-coordinate into the equation and solve for y.
2nd Make a table of values. Choose points on both sides of the vertex
3rd Graph the parabola
17.0, 21.0/A-SSe.3a,b, F-IF.7a
Practice Graph the functions:
44) ( ) 2 2g x x x= − −
45) 2 2y x x= − +
46) ( ) 2 4f x x= − +
47) y = 3x2
For #48 & 49, find the domain and range of the relation Is each relation a function?
48) 0, 2( ), 3,1( ), 3,−1( ), 5,3( ){ }
49)
50) What is the range of the function f x( ) = −2x3 + 3 when the domain is −3, 0, 2, 4{ }?
x y 2 5 3 – 5 4 5 5 – 5
( )( ) ( ) ( )
( ) ( )
2
2
2 4 1
1 2 1 4 1 1
2 1 4 13
f x x x
f
= + −
− = − + − −
= + − −= −
( )vertex 1, 3∴ − −
x y x x= + −2 4 12 y 1 0 -1 -3 -2
( )
242 21
bxa
x
x
=−
=−
= −
HUSD Middle School Algebra I Final Assessment Study Guide
Page 9 of 10 MDC@ACOE (HUSD) 04/28/13
Algebra 1 Study Guide Answer Key
1) ( ) ( )6,0 ; 0,2 16) ( )3, 2− 31) 19, 19x x=− = 2) ( ) ( )3,0 ; 0, 1− 17) ( )9, 13− 32)
x = 2
3, x = 1
3) ( )4, 0, 3m = − 18) ( )2, 3− 33) No Solution
4) ( )1 , 0,23
m = − 19) 7 5,4 2
⎛ ⎞⎜ ⎟⎝ ⎠
34)
x = −3± 29
2
5)
20) ( )10, 4 35)
x = −1, x = 5
3
6)
-5 -4 -3 -2 -1 0 1 2 3 4 5
-3
-2
-1
1
2
3
21) ( )300, 100
There are 300 nickels and 100 dimes.
36) 7 1,2 2
x x=− =
7) 12
m = − 22) 4, − 3( ) 37) 1 6x = ±
8) 2m = − 23) 6 44 2p q p 38) 5, 2−
9) No slope, undefined slope
24) 212w x wx 39) 9 32
z z⎧ ⎫− ≤ ≤⎨ ⎬⎩ ⎭
10) 1x = − 25) 2 55rr
40) 5 or 42
x x x⎧ ⎫< − >⎨ ⎬⎩ ⎭
11) 5y = 26) 9 2 41) { }5 or 5y y y≤ − ≥ 12) 3y x= − + 27) 13 or 25x x= = 42) { }1 4x x− ≤ < 13) y = 2x +1 28) 45
4z = 43) 18 or 18x x=− =
14) 13
m = − 29) 0, 1, or 2x x x= = − = −
15) 3m = 30) 5, 9x x=− =
HUSD Middle School Algebra I Final Assessment Study Guide
Page 10 of 10 MDC@ACOE (HUSD) 04/28/13
44) 45) 46) 47) 48) Domain {0, 3, 5} Range {–1, 1, 2, 3} No it is not a function because where x = 3, the equation has two values of y. 49) Domain {2, 3, 4, 5} Range {–5, 5} Yes it is a function. 50) When the domain is {–3, 0, 2, 4} the range is {57, 3. –13, –125}
-5 -4 -3 -2 -1 0 1 2 3 4 5
-4
-3
-2
-1
1
2
3
4