Algebra I SOL Topics and Formulas

Expressions Follow order of operations PEMDAS Combine like terms (same variables with same exponents) Input Output ( )x∫ = y (output)

X= input (replace every x in the equation with the number in ( )

2(5) 5x x= +∫ Plug in value for x 25 5(5)50

yy= +

=

f(7) = 3x – 5 y= 3(7) – 5 y=16 Equations, Inequalities In calculator Go to Menu Equation Solver Type in the equation exactly how you see it ( include ( ) if needed) Hit exe TWICE!! Inequalities:

Use the same steps as solving equations. **(if multiplying or dividing by a negative FLIP the sign in your answer!!!)

2 147

xx− ≥

≤ −

Function, Patterns Function: x does not repeat (domain is all different numbers) Graphs use vertical line test (can only cross at one point)

EX1: {(3, -4), (3, 5), (4, 6)} EX2: {(7,1), (8,1), (9,1)} EX3 EX4 not a function – the x’s repeat function EX5: EX6: Functions Not a Function – Not a function D:{3,5} R: {4, 10} Not a function

1 2 3

4 5 6

2 5

8 9 10

6 12 1

1 5 5

4 4 10

3 3 5

Dividing by a negative…. flip the sign!

f(x)=abs (x+2)

-9 -8 -7 -6 -5 -4 -3 -2 -1 1 2 3 4 5 6 7 8 9

-9-8-7-6-5-4-3-2-1

123456789

x

y

x^2/9+y^2/25=1

-9 -8 -7 -6 -5 -4 -3 -2 -1 1 2 3 4 5 6 7 8 9

-9-8-7-6-5-4-3-2-1

123456789

x

y

x=3

-9 -8 -7 -6 -5 -4 -3 -2 -1 1 2 3 4 5 6 7 8 9

-9-8-7-6-5-4-3-2-1

123456789

x

y

function not a function – use the vertical line test not a function – use the vertical line test D = { all real numbers} R = { all real numbers greater than 0} Linear Equations, X, Y, Intercepts y=mx+b m=slope b=y-intercept Ax + By = C standard form Horizontal line (HOY) slope = 0 equation starts with y Ex. y=4 Vertical line (VUX) slope = undefined equations starts with x Ex. x=4 To fine the x and y intercept from standard form: Ax + By = C 3x-4y=12 x-intercept y-intercept Cover up the y value Cover up the x value Solve for x Solve for y 3x=12 -4y=12 X=4 y=-3 On Calculator: Solve for y=mx+b Menu Graph Type in equation F6 Draw F5 GSOLV F1 ROOT for the x-intercept F4 YICPT for the y-intercept Slope

Slope= 1 2

1 2

y yx x−

− m=

AB−

Positive Negative Zero Undefined

x

y

x

y

x

y

x

y

1 2 3 4 5 6 7 8 9 10–1–2–3–4–5–6–7–8–9–10 x

12345678910

–1–2–3–4–5–6–7–8–9–10

y

1 2 3 4 5 6 7 8 9 10–1–2–3–4–5–6–7–8–9–10 x

12345678910

–1–2–3–4–5–6–7–8–9–10

y

1 2 3 4 5 6 7 8 9 10–1–2–3–4–5–6–7–8–9–10 x

12345678910

–1–2–3–4–5–6–7–8–9–10

y

Write Linear Equations Y=mx+b slope intercept (where it crosses the y axis)

(m is your slope (riserun

) , b is the y-intercept)

Ax+By=C Standard Form -A, B, and C cannot be fractions

-A cannot be negative - GCF has to be 1

b (y-intercept) Writing an equation of a line from a graph:

1. Find b (y-intercept) 2. Find m (slope) 3. Count up and over to the other point. 𝑟𝑖𝑠𝑒𝑟𝑢𝑛 𝑦 = !!

!𝑥 + 1

4. Plug in m and b into the equation y=mx+b Writing an equation of a line from a point and a slope: (5,-2) m=!!

!

1. Label your point and your slope x=5 y= -2 m=−35 2. y= mx+b plug in the values for y, x, and m y=mx+b

-2=−35 (5) +b 3. Solve for b -2= -3+b

1=b 4. y=mx+b plug in the values for m, and b y=−35 x +1

Writing an equation from two points: (5, -2) (1,0) In Calculator

1. Menu 2. Stats 3. Place x values in L1 4. Place y values in L2 5. F1 Graph 6. F1 Graph 7. F2 x 8. F1 ax+b 9. Replace the a value for a and the b value for b

Systems of Equations: 2 equations, 2 variables, 2 answers

If the 2 lines intersect the solution is (x,y) *you only see two lines touch and make an X*

Slope (m) are different Y-int. (b) are different If they do not cross then they are parallel, no solution *you only see two lines that do NOT touch*

Slope (m) are the same Y-int. (b) are different

1 2 3 4 5 6 7 8 9 10–1–2–3–4–5–6–7–8–9–10 x

12345678910

–1–2–3–4–5–6–7–8–9–10

y

Rise

Run

1 2 3 4 5 6 7 8 9 10–1–2–3–4–5–6–7–8–9–10 x

12345678910

–1–2–3–4–5–6–7–8–9–10

y

If they are the same line, infinitely many solutions. *you only see one line*

Slope (m) are the same Y-int (b) are the same

On Calculator: Set each formula in standard form Ax By C+ = . Set Each formula in slope intercept form y=mx+b Menu Menu EQUA Graph

F1: Simultaneous Type in both equations F1: 2 unknowns (x and y) F6: Draw Type in A, B, and C for each equation F5: ISCT (they intercept, 1 solution (x,y)) F1: Solve You see 2 lines DO NOT touch, no solution You only see ONE line, infinitely many **If you get MA ERROR you must graph your lines, your answer will be Infinitely many or No Solution. Monomials, Exponents, Scientific Notation Product of Powers-same base Power of a Power

nmnm aaa +=⋅ nmnm aa ⋅=)( Power of a Product Quotient of a Power

mmm baab =)( nmn

m

aaa −=

Power of a Quotient Negative Exponents

x

xx

ba

ba

=⎟⎠

⎞⎜⎝

⎛ a

a

xx 1

=− or

aa x

x=

−

1

0x is always equal to 1 Scientific Notation 2.53 x 510 = 253,000

3.06 x 510− = 0.0000306 Polynomials Add/Subtract like terms

(The same variables with the same exact exponents) 2 43 4 2xy xy x y+ −

2 413xy x y+12 − 2 43 16 15xy xy x y+ −

Multiply/Distribute, FOIL (First Outer Inner Last) ( 5)( 8)x x+ − 2( 2)( 3 8)x x x+ − +

2 8x x− 3 2 8x x x−3 +

5 40x+ − 22x x+ −6 +16

3 2 16x x x− + 2 +

(x+2)(2x-3)

2x2+x-6

Rules for Factoring: 1. Look for a GCF

a. If you find one take it out. In Calculator:

i. Menu ii. Run

iii. Optn iv. F6 v. F4 num

vi. F6 vii. F2 GCD

Type in your #’s with a comma in between. Ex.1 (5, -10, -20)

Ans: 5

Ex 2) 3 2

2

8 568 ( 7)a x aa ax

−

−

Ex 3) 215

5 (3 )a aa a

−5

−1

** When taking out a variable as the GCF, take out the one with the lowest exponent.

Ex: 4 2 2 3 3 2 4

2 2 2 3( )ab x a b x a b xab x b x ab a x

− +

− +

2. If there are 4 terms: a. Find the GCF of ALL 4 terms

4 3 2

3 2

8 64 8(8 64 8)r r r rr r r r

− + +

− + +

b. Break your four terms into two groups

3 2(8 64 )( 8)r r r− +

c. Take out the GCF of the 1st ( ) 3 2

2

(8 64 )8 (r rr r

−

−8)

2 3 40x x− −

d. Take out the GCF of the 2nd ( ) *** you MUST make the ( ) match the first one!!!

( 8)1(rr+

− −8)

** Notice that our ( ) match!!

e. Bring the GCF from step C and D into one set of ( ) and bring one of the matching ones down 2(8 )( 8)r r−1 −

f. If you have a GCF in step a bring that out in front. 2(8 )( 8)r r r−1 −

3. When you have 3 terms. In Calculator:

a. Take out the GCF b. Set the equation equal to zero 2 0Ax Bx C+ + = c. Menu d. EQUA e. F2 Polynomial f. F1: 2nd degree (highest exponent after the GCF) g. Type in the values for A, B, and C (if there is no A, B, or C in your equation type in 0) h. F1: Solve

These are your roots, solutions, zeros to get the factors: place in (x___) with the opposite value.

2 3 4x x− −

[ ][ ]1

4

x

x

− The factors will be (x+1)(x-4)

24x x−5 −6 [ ][ ]3 / 4

2

x

x

− The factors will be (4x+3)(x-2)

***Remember you can check all your work by distributing or doing the box method. Radicals Simplifying Radicals To simplify, use n n nab a b= • or break out into prime factors looking for the same repeated factors (2 or 3 or 4 of a kind—depending on the index).

2

27 3 3 3 3 3of a kind

= =g gEF 354 = 3 3 3 2g g g 33 2=

Comes out of the radical

Simplifying Radicals with variables:

To turn a root into a factor: Take the opposite sign and put in (x __) after x

If you have a fraction as a root: the denominator goes in front of (_x) and the numerator takes the opposite sign and goes after x

1 2 3 4 5 6 7 8 9 10–1–2–3–4–5–6–7–8–9–10 x

6

12

18

24

30

–6

–12

–18

–24

–30

y

2325 5 5a pair

x x x x x= =gEF 3 35 7 2 216 2 2 2 2 2 2a b a a a a a b b b b b b b ab a b= =g g g g g g g g g g g g g g g

Solve Quadratic Equation Set trinomial =0 Factor Polynomial like normal (using the calculator) Set each factor (set of parenthesis) equal to zero Solve for the variable You should have two answers!

2 6 27 0( 9)( 3) 0( 9) 09

( 3) 03

x xx xxxxx

− − =

− + =

− =

=

+ =

= −

Vertex: (3,-36)

Domain: R or (-∞,∞) Vertex Range: 𝑦 ≥ −36 (minimum value opens up)

Vertex max opens down

{ }

2

2

120 120 ( 3)( 4)

3, 4

y x xx xx x

x

= − − +

= − − +

= − +

= −

Vertex: (-0.5,12.25) Domain: R or (-∞,∞) Range: 𝑦 ≤ 12.25 Using the calculator: Set equation equal to zero Roots: Y-Intercept:

Menu Menu Graph Graph Type in equation Type in equation F6: Draw F6: Draw F5: GSOLV F5: GSOLV F1: Root F4: Y-ICPT Use the side arrows to go from one root to the other

Vertex: Domain: Always all real numbers or (-∞,∞)

Menu Graph Range: Type in equations Take the y-value from the vertex insert it into equation

Factors

Roots, Zeros, Solutions

Factors

Roots, Zeros, Solutions (Where the graph crosses the x-axis)

1 2 3 4 5 6 7 8 9 10–1–2–3–4–5–6–7–8–9–10 x

6

12

18

24

30

–6

–12

–18

–24

–30

y

F6 Draw Graph opens up use 𝑦 ≥ F5 GSOLV Graph opens down use 𝑦 ≤ F2 for MAX (opens down)

F3 for MIN (opens up) Line of Best Fit On calculator: Menu STAT Type all values of X in list 1 Type all values of Y in list 2 F1: GPH1 F1:GPH1 F1:CALC F2: X F1: ax+b Replace a and b with the data given if it is linear

• Look at the scatterplot graph, decide which model is most reasonable (linear) • Calculate the formula using y=ax+b plugging in a, and b

Curve Best Fit On calculator: Menu STAT Type all values of X in list 1 Type all values of Y in list 2 F1: GPH1 F1:GPH1 F1:CALC F2: 2x F6:draw

Look at the scatterplot graph, decide which model is most reasonable (quadratic)

Write Equation using 2y Ax Bx C= + + Plug a, b, and c back into the equation. Round to the second decimal place.

Direct Variation • Equation: y = kx (k is the constant of variation); • graph is a line thru the origin • Solve first equation for k • substitute k into another equation and solve for the unknown variable

Example 1: If y varies directly as x and y is 6 when x is 18, find y when x is 24.

Set up to solve for k: 6 1813

y kxk

k

=

=

=

Then plug k into the formula and find the missing variable

x

y

1 (24)38

y kx

y

y

=

=

=

X Y Direct Variation multiple DIAGONLY across.

II Inverse Variation

• Example – the speed of a car and the time it takes to reach the destination

• Equation: xy k= or kyx

= ( k is the constant variation)

• graph is a hyperbola in opposite quadrants (Quad I & III or Quad II and IV) • To solve find k and substitute it and remaining numbers into eqn. again.

Example 2: If y varies inversely as x and y = 10 when x = 2, find y when x = 6.

If (2)(10)20

xy kk

k

=

=

=

Find y now

6 20206103

xy ky

y

y

=

=

=

=

X Y Inverse you multiply INLINE (across)

2 10

6 y

Statistics Mean- averages (add up all the data and then divide by # of objects) Mode- Most Median- middle # put in order smallest to biggest Range- subtract the lowest value from the highest value

• Standard Deviation, Variance, and Z-score

o Calculator Instructions: Menu Stat F4 Del A (you may need to hit F6 to see Del A on calc) Type in all data into L1 F6 if you don’t see CALC F2 CALC F1 1 VAR σx standard deviation

18 6

24 y

x

y

x̅ is the mean (µ) n is the total number of terms

Variance= (Standard Deviation)2

(std.dev × std.dev) Z-Score

o A positive z-score: above the mean o A negative z-score: below the mean o A z-score of zero: at the mean

o Formula:

Plug in the vaules you have and solve for the unknow Empirical Rule

In a normal distribution with mean µ and standard deviationσ : 68% of the data fall within 1σ of the mean µ 95% of the data fall within 2σ of the mean µ 99.7% of the data fall within 3σ of the mean µ

Creating a Box and Whisker

Calculator Instructions: Menu Stat F4 Del A (you may need to hit F6 to see Del A on calc) Type in all data into L1 F6 if you don’t see CALC F2 CALC F1 1 VAR Scroll down minX is your Lower Extreme Q1 is your lower quartile Med is your Median Q3 is your upper quartile maxX is your Upper Extreme