husd middle school algebra i final assessment study guide · 2013-05-15 · husd middle school...

HUSD Middle School Algebra I Final Assessment Study Guide

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

+ = −+ = −

= −= −



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

=

− −=− − −

=



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.



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

=+ =

+ − =− =

==



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



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



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− < <



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

=−

=−

= −



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=− =



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

husd middle school algebra i final assessment study guide · 2013-05-15 · husd middle school...

Documents