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

Menu

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

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

k

1

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3

8

y kx

y

y

xy k

y

k

x

k

(2)(10)

20

xy k

k

k

6

20

20

6

10

3

xy k

y

y

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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Figure

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References

Related subjects : Algebra I