Henrico-Algebra I SOL Topics and Formulas.docx

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Algebra I SOL Topics and Formulas

Expressions

Follow order of operations PEMDAS

Combine like terms (same variables with same exponents)

Input

Output

= y (output)

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

Plug in value for x

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!!!)

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 Dividing by a

negative…. flip the sign!

8 9 1 2

5 4

5 6 1

2 3 4

4 10 3

3 5 6

1 2 1

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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

Graph

Type in equation

F6 Draw

F5 GSOLV

F1 ROOT for the x-intercept

F4 YICPT for the y-intercept

Slope

Slope= m=

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

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

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

.

rise

run

y=−3

5 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=−3

5

1. Label your point and your slope

x=5 y= -2 m=

−₅3

2.

y= mx+b plug in the values for y, x, and m y=mx+b

-2=−3

5 (5) +b

3.

Solve for b -2= -3+b

1=b

4. y=mx+b plug in the values for m, and b

y=

−3

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

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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

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 . 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

Power of a Product Quotient of a Power

Power of a Quotient Negative Exponents

or

is always equal to 1

Scientific Notation 2.53 x = 253,000 3.06 x = 0.0000306

Polynomials

Add/Subtract like terms

(The same variables with the same exact exponents)

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

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Multiply/Distribute, FOIL (First Outer Inner Last)

(x+2)(2x-3)

2x

2

_+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)

Ex 3)

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

Ex:

2. If there are 4 terms:

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b. Break your four terms into two groups

c.

Take out the GCF of the 1st_{( )}

d.

Take out the GCF of the 2nd_{( ) *** you MUST make the ( ) match the first one!!!}

** 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

f. If you have a GCF in step a bring that out in front.

3. When you have 3 terms. In Calculator:

a. Take out the GCF

b. Set the equation equal to zero 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.

The factors will be (x+1)(x-4)

The factors will be (4x+3)(x-2)

***Remember you can check all your work by distributing or doing the box method.

Radicals

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

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Simplifying Radicals

To simplify, use

or break out into prime factors looking for the same repeated factors (2 or

3 or 4 of a kind—depending on the index).

Comes out of the radical Simplifying Radicals with variables:

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!

Vertex: (3,-36)

Domain: R or (-∞ , ∞¿ Vertex

Range: y ≥−36 (minimum value opens up)

Vertex

max opens down

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

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

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

n

_ab

_

n

_a

_

n

_b

2

27 3 3 3 3 3

of a kind



_

_{ }



₃

54 

2

3

25

5

a pair

x



_

x x

_



x x

Factors

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

Factors

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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

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 bwith 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:

F6:draw

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

Write Equation using

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

x

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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:

Then plug k into the formula and find the missing variable

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:

or

( 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

Find y now

X Y

Inverse you multiply INLINE (across)

2

10

6 y

x y

6 18

1

3 y kx

k



1 (24)

3

8 y kx

y



xy k



y



k

_x

_k

(2)(10)

20 xy k

k



6

20

6

10

3 xy k

y



18

6

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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 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 and standard deviation : 68% of the data fall within 1 of the mean

95% of the data fall within 2 of the mean 99.7% of the data fall within 3 of the mean

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

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Henrico-Algebra I SOL Topics and Formulas.docx

Full text

Algebra I SOL Topics and Formulas

Expressions

Input

Output

Equations, Inequalities

Function, Patterns

Linear Equations, X, Y, Intercepts

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

Write Linear Equations

1. Find b (y-intercept)

2. Find m (slope)

3.

.

4. Plug in m and b into the equation y=mx+b

1. Label your point and your slope

x=5 y= -2 m=

2.

3.

4. y=mx+b plug in the values for m, and b

y=

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

Monomials, Exponents, Scientific Notation

is always equal to 1

Polynomials

(x+2)(2x-3)

2x

+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

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

Ex:

2. If there are 4 terms:

b. Break your four terms into two groups

c.

d.

***Remember you can check all your work by distributing or doing the box method.

Radicals

To simplify, use

or break out into prime factors looking for the same repeated factors (2 or

3 or 4 of a kind—depending on the index).

Solve Quadratic Equation

n

ab

_+x-6

_ab

_

_a

_

_b

_

_{ }

₃

_

_