Algebra and Functions

Factoring Polynomials w/cubes

4xᶟ - 9x = x(4x² - 9) = x (2x² - 3²) = x(2x² - 3)(2x + 3).

Linear Equation: Slope Intercept

Equation: y = mx + b ; "m" is slope, and "b" is y-intercept.

Evaluate an Expression

replace variables with values: Ex: 3x-y/z ; x = 3, y = 8, z = 4. 3(3)-(8)/(4) = 9 - 8/4 = 9 - 2 = 7.

Quadratic Equation using FOIL method

standard form is ax² + bx + c = 0. When graphed, a quadratic equation will form a parabola that opens up or down, but never sideways. When having an expression in the form (a + b) (c + d) we can distribute using F = first, O = outer, I = inner, L = last. Ex: (x-2)(x+1) = x² + x - 2x - 2 = x² - x -2. FOIL is good method to check if we have factored correctly. Factoring & distributing are algorithmic opposites.

Trinomial

sum, difference, multiplication or division of three terms: 3x + 5y² - 3; x² + 6x + 9; 4xy² - 2x + 3y, etc.

Binomial

sum, difference, multiplication or division of two terms: 5x - 1; 3x + 5; x² - y²; etc.

Input

the domain, x-values - independent, non-repeating quantity.

Solving systems of equations

the first goal is to eliminate ONE variable. Systems of equations are 2 or more equations that relate to one another. on CSET, it could ask which step would be needed to solve instead of, or in addition to provide an answer. On multiple choice, you could plug the answer choices on the variables and see which one solves out.

Linear Equation: Point-Slope form

used when you know the coordinates of one point on the line and slope. The equation is: y-y1 = m(x - x1)

Graphing Inequalities

Begin by graphing the line as you normally would. Then determine shading based on the <,>,≤, or ≥ sign. Ex: graph y > - 2x - 1 *If "y" is greater than the x-value given on the inequality, shade above. If "y" is less than, shade below it. "Greater than" or "equal to" have solid lines.

How to graph a quadratic equation

Ex: x² + 5x + 6 Vertex/Axis of Symmetry: (-5/2, -1/4) y-intercept (b) = (0)² + 5(0) + 6 y = 0 + 0 + 6 y = 6 = (0,6) (use quadratic equation to determine x-intercepts.) For this equation it is, (-2,0) and (-3,0). So mark the graph with all its intercepts and determine whether the parabola opens upwards or downwards. In this case it opens upwards.

Graphing Linear Inequalities: false

Ex: y ≥ -3x + 2 (you could forget inequality symbol and turn it into normal equation): y = -3x + 2. y-intercept: 2 ; (0,2). Slope: -3/1. Test (0,0) any points can be tested, as long as they are not on the line: 0 ≥ -3(0) + 2 = 0 ≥ 0 + 2 = 0 ≥ 2. False! zero is not greater than or equal to 2. Any points below (0,0) are false, so shade above line.

Solving systems of equations: addition/substitution (or elimination) method continued

Next step, plug in the new variable value into either of the equations, and figure out value for 2nd variable, in this case "y" : 2x + y = 3 2 (-3) + y = 3 -6 + (6) + y = 3 + 6 y = 9.

Relations & Functions

https://www.montereyinstitute.org/courses/Algebra1/COURSE_TEXT_RESOURCE/U03_L2_T1_text_final.html

Inequality Symbols

if the value is NOT part of the solution set, use: <, > greater than or less than signs. If the value IS part of the solution set use, ≤, ≥ greater/less than or equal to signs.

Output

the range, y-values - dependent quantity. The value of the output depends on the value of the input. Ex: when tossing a ball, time is the input and height is the output.

Find missing coordinate using slope

(6,0) and (7,d) with slope of -8. What is the value for d? Slope = change in y/change in x -8 = d - 0/7 - 6 -8 = d

Polynomial Name by Degree

0 = Constant, e.g. 7 (lines are straight). 1 = Linear, e.g. 4x + 3 (lines are diagonal). 2 = Quadratic, e.g. x²−3x+2 (lines are slope/hilly). 3 = Cubic, e.g. 2x6^3−5x² (lines are curvy). 4 = Quartic, e.g. x^4+3x−2 (lines have larger curves).

Monomial

1 term: 3xy² , 3x, 7y², 5, etc. There is operation involved also, like: 3x - 2x, 5 + 1, 24/4, etc.

Divide Polynomials (Rules)

1) divide 1st term of numberator by 1st term of the denominator, and put it as an answer. 2) multiply the denominator by that answer and put it below the numerator. 3) subtract to create a new polynomial. 4) when having remainders, put them divided by the same denominator. 5) if missing terms, leave gaps or put zero. 6) check by multiplying answer by the divisor.

Divide Monomials (simplify rational expression)

1. 28b^6/7b = 28/7 * b^6/b^1 = 4b^5 2. 6x³/5 * 2/3x = 12x³/15x^1 = 4/5 x² 3. 2x⁴/7 ÷ 5x⁴/4 = 2x⁴/7 * 4/5x⁴ = 8x⁴/35x⁴ = 8/35

Factoring Polynomials w/no GCF

10x² - 13x - 3 = (2x - 3)(5x - 1), because there are no GCF that give product of -3 equal to sum of -13, you can skip this step and find products that only equal to -3.

Graphing

Composed for 4 quadrants, x and y axis. Also known as horizontal and vertical axis. Two points determine a line, slope, y-intercept, and slope-intercept, and form an equation.

Independent variable

plotted on the x-axis

Dependent variable

plotted on the y-axis

Zero-Factor Property

states if the product of two factors is 0, then one or both of the factors must equal to 0. It also determines where the parabolas will cross the x-axis for the quadratic equations.

Factoring Polynomial

4x⁴y - 8x³y - 2x² (GCF: 2x²) 2x² (4x⁴y/2x²) - 2x² (8x³y/2x²) - 2x² (2x²/2x²) 2x² (2x²y) - 2x² (4xy) - 2x² (1) 2x² (2x²y - 4xy - 1).

Writing an inequality sample

Debbie has at most $60 to spend on clothes. She wants to buy a pair of jeans for $22 and spend the rest on t-shirts. Each shirt cost $8. Let x = # of t-shirts. Since jeans + shirt could equal $60 or less than $60, jeans + shirt ≤ $60. Then translates to: 22 + 8x ≤ $60.

Direct Proportion

If one variable is always the product of the other. x and y are directly proportional if the ratio yx is constant.

Factoring Polynomials example

x² + 8x + 15 (figure the GCF, 2 numbers that equal product of 15 but adds to 8) (x + 3)(x + 5) is the result.

Linear Polynomial Sample

y = 2x + 1 2x + 1 = 0 2x = -1 x = -1/2 Graph the points (-1/2, 0).

Ratio

A comparison of two quantities by division; it is a fixed relationship between two quantities. Ex: for every bag of cookies, you have one chocolate chip for every two oatmeal cookies. Then keeping that same ratio, how many oatmeal cookies when having 2 chocolate chips? 1 : 2, 2 : x? OR 1/2 = 2/x ? 4 oatmeal cookies. 1/2 : 2/4.

Linear Equation

A linear equation is an equation that describes a straight line on a graph. A linear equation stands like this: Ax + By = C. A, B, C are coefficients and x, y are variables and are points on the graph. There are other ways the equation can be written. - Plug A, B, & C numbers onto the equation like this: 2x + 3y = 7; x + 7y = 1; 3x - y = 1; e.g.

Solving systems of equations: graphing method

CSET may ask you to graph systems of equations. Graph the following systems of equations: 2x + y = 3 2x + 2y = 2. 1. Start solving or manipulating them in slope-intercept form y = mx+b. 2x -(2x) + y = 3 - 2x 2x - (2x) + 2y = 2 - 2x y = -2x + 3 (0,3) slope : -2/1 2y/2 = -2x/2 + 2/2 y = -x + 1 (0,1) slope: -1/1 2. Then graph. 3. Draw lines and intersect them. 4. Where the lines intersect, is the solution.

Example #2 of Ratio

In a classroom with 15 total kids there are 3 kids with blue eyes, 8 kids with brown eyes, and 4 kids with green eyes. What is the ratio of blue eyed kids to kids in the class? 3:15 or 3/15.

Line Graph

Straight and extends infinitely in both directions.

Simplify expression (factoring)

3x + 3y + 3z + 2 (x + y+ z) = 3 (x + y + z) + 2 (x + y + z) = 5 (x + y + z) .

Linear factor

two expressions whose product is the quadratic equation: (4x + 2)(3x - 1) = 12x² + 2x - 2

Quadratic equation for x² + 5x + 6

x = −5 ± √(5² − 4(1)(6))/2(1) x = -5 ± √(25 − 4(6))/2 x = -5 ± √(25 − 24)/2 x = -5 ± √(1)/2 x = -5 ± 1/2 x = -4/2 = -2 or x = -6/2 = -3 (-2,0) and (-3,0)

Inversely Proportion

When one variable increases the other decreases in proportion so that the product is unchanged.

Equation

a mathematical statement that two expressions are equal. Something is set to equal something else, can solve for "x." x + 2 = 3 x + 2-2 = 3-2 x = 1 1 + 2 = 3 3 = 3 .

Inequality Example

-3x + 3 ≥ 12 -3x ≥ 9 -x ≥ -3 (flip the inequality sign) = x ≤ -3

Polynomial

4xy² + 3x - 5; "many terms" it can have addition, subtraction, multiplication, and division. They can have no variable, consists one term only, and can have more than one variable. But no division by a variable: 2/x, or 2/x+2, 1/x, etc.

Point Slope Problem Sample

Equation: y - y1 = m(x - x1) y1 and x1: the coordinates of the points you know. m = slope. x and y: variables. Problem: Graph a line that passes through the coordinate (2,2) and has a slope of 3/2. Write the equation in slope-intercept form. Step 1: First plot the point (2,2) and rise 3 and run 2 for 3/2. Draw line between two points. Step 2: use equation y = mx + b; m = 3/2, and b = -1. Therefore, y = 3/2x - 1. See link above for more examples.

How to determine a linear equation from table

Ex: Slope m = y/x = -5/1 = -5; y-intercept = (0,15) when filling up the xy table backwards from 2 and 5: 1 | 10, and 0 | 15. x | y 2 | 5 3 | 0 4 | -5 5 | -10 Linear equation: y = -5 x + 15

Linear Equation Problem Sample

Graph linear equation: 2x + y = 2. Step 1: Start with zero 2(0) + y = 2, then y = 2. Step 2: 2x + (0) =2, then 2x = 2, as a result x = 1. Step 3: Graph the points (1,0) and (0,2). Step 4: Draw a straight line through the points. Step 5: Check answer by plugging other numbers on the variables. http://www.ducksters.com/kidsmath/linear_equations.php

Slope Intercept Problem Sample

Graph the equation y = 1/2 x + 1. m = slope = 1/2 b = intercept = 1. See more examples: http://www.ducksters.com/kidsmath/linear_equations_slope_forms.php

Example of Direct Proportion

Terrell bought 9 postcards after 3 days of his trip. How many postcards will he have after four days? 9 postcards/3 days = x postcards /4 days x = 12 postcards.

Simplify expression (combining like terms)

a "like" term has the same variable (or lack thereof) to the same power: 2x² + 3x + 9 -7x + x² + 2x - 5 = 3x² - 2x + 4 .

Variable

a quantity that may change within a mathematical problem. It typically letters (x,y,z) or symbols that represent an unknown number or set of numbers.

Equalities

a relationship between two quantities or math expressions, produce the same value or mathematical object. All expressions must connect with an equal (=) sign.

Rise/Run

another form of determining the slope. The rise is the distance the line travels up or down, the run is the distance the line travels from left or right.

Word problems using related quatities

creates a ratio between the numbers of various items in a set.

Solving systems of equations: addition/substitution (or elimination) method

Ex: solve for x and y in the following system: 2x + y = 3 3x + 2y = 9 want to get the "y" variable equal to "2y" term, so multiply by 2. 2 (2x + y) = (3) 2 4x + 2y = 6 -(3x + 2y = 9) = New system equation. Now we subtract, one equation from another equation. "2y" term is eliminated and left w/single variable. x + 0 = -3; Answer: x= - 3.

Relation

Domain. Input of numbers.

Graphing Linear Inequalities: true

Ex: x-y > -2 x - (x) - y > -2 - (x) -y/-1 > -x-2/-1 y < x + 2 y-int. : 2; (0,2) Slope: 1/1. Test (0,0): 0 < 0 + 2 = 0 < 2; true! If true, shade below line.

Simplify expression (distribution)

2 (2x + 3x + 9) = 4x + 6x + 18 = 10x + 18 . OR 2 (5x + 9) = 10x + 18 .

Graphing Linear Inequalities: determine x and y intercepts

2x + 5y ≤ 10 or, 2x + 5y = 10. Test (0,0): 2x + 5(0) = 10 2x + 0 = 10 2x = 10 (divide both sides by 2) x = 5; x-intercept: (5,0). 2(0) + 5y = 10 0 + 5y = 10 5y = 10 (divide both sides by 5) y = 2; y-intercept: (0,2). Test (0,0) 2(0) + 5(0) ≤ 10 0 + 0 ≤ 10 true!

Solving systems of equations: substitution method

2x + y = 3 3x + 2y = 9 "Solve for y" or "solve for y in terms of x": 1. 2x + y = 3 2x - 2x + y = 3 - 2x; (the 2x cancels out) y = -2x + 3. Next, plug in the new "y" variable equation onto the other equation, NOT the one you solved: 3 x + 2 (-2x + 3) = 9. Now solve it. 3x - 4x + 6 = 9 -x + 6 = 9 - x + 6 - 6 = 9 - 6 - x/-1 = 3/-1 x = - 3. This can then be substituted to figure out for y value.

If the number 360 is written as a product of its prime factors in the form aᶟb²c, what is the numerical value a + b + c?

360 = (10)(36), (12)(30), (360)(1) etc. (10)(36) = (2)(5)(6)(6) and continue breaking down until you get prime factors. (2)(5)(6)(6) = (2)(5)(2)(3)(2)(3) or 2ᶟ x 3² x 5 Next, add the prime factor numbers 2 + 3 + 5 = 10.

Writing an expression sample

Laila tells Julius to pick a number between 1-10. "Add 3 to your number, and multiply the sum by 5." She tells him. "Now take that number and subtract 7 from it and tell me the new number." "23" Julius exclaims. Write an expression that records the operations Julius used. (n = Julius number) 5(n + 3)-7 = 23 5n + 15 = 30 5n = 15 n = 3 (Julius original number.)

Perpendicular Lines

Line that crosses another line at a 90° angle. Each corner meets at a 90° angle. This is also known as a right angle. Perpendicular lines can face any direction. They do NOT need to be standing straight up from the bottom or side of the page. They just have to be 90° to each other.

Plotting fractions on graphs

Move to the right the number of spaces equal to the denominator, and move up or down (depending if its positive or negative variable) the number of spaces equal to the numerator.

Function, f(x)

Range. Output of numbers. It describes the relationship between two sets of data. Functions without an exponent generally produces straight lines (y = x). Functions with an exponent usually produce curved lines (y = x²). The rate of increase is not constant.{(1,2) (1,4) (2,-3)} not a function because the x-values repeat.

Quadratic Equation Problem Sample

Solve 5x² + 6x + 1 = 0. use quadratic formula (as seen on picture). a = 5, b = 6, c = 1: x = −6 ± √(62 − 4×5×1)/2×5 x = −6 ± √(36 − 20)/10 x = −6 ± √(16)/10 x = −6 ± 4/10 x = −0.2 or −1 Check answers: 5×(−0.2)² + 6×(−0.2) + 1 = 5×(0.04) + 6×(−0.2) + 1 = 0.2 − 1.2 + 1 = 0 5×(−1)² + 6×(−1) + 1 = 5×(1) + 6×(−1) + 1 = 5 − 6 + 1 = 0

Expression

Something that is not set to equal to something else. You can't solve for x, but can evaluate its behavior and represent relationships or pattern. Ex: 1 blue cube, 2 yellow cubes, 3 green cubes: x + 2x + 3x = total number of cubes.

Completing the square formula

a (x + b/2a)² + c - b²/4a = 0 Ex: x² - 4x - 13 = 0; a = 1, b = -4, c = -13 1 (x - 4/2*1)² - 13 - (-4)²/4*1 = 0 1 (x-2)² - 13 - (16)/4 = 1 (x-2)² - 13 - 4 = (x-2)² - 17 = √(x-2)² = √17 = x - 2 + 2 = + or - √17 + 2 = x = 6.12 or - 2.12

Inequalities

are a set of five symbols used to demonstrate instances where one value is not the same as another value: <, >, ≤, ≥, and ≠. They can be solved using the same basic techniques used to solve equations, but remember to always reverse the sign whenever you multiply or divide by a negative. Ex: Solve for x: 5 > -3x-1 = 5 (+1) > -3x -1 (+1) = 6/-3 > -3x/-3 = -2 < x

Proportions

based on setting that two ratios equal to one another. Rule of three's can be helpful. It tells when we know two ratios are proportionate, and we have a 3 of 4 quantities we can solve for the fourth. Typically cross-multiplication is used: 1/2 = 2/x; "x" is the 4th quantity. 1 * x = 2 * 2 x = 4. 1/2 = 2/4 1/2 = 1/2 .

systems of linear inequalities

consists of 2 variables, and 2 linear inequalities. Always deal with one quadrant, and has more than 1 solution.

Slope

describes how steep a straight line is. It is sometimes called the gradient. The slope is defined as the "change in y over the change in x" of a line. Formula: (y2 - y1/x2 - x1) or another way: m = y/x.

Equivalent Expression

equivalent expressions are the same, even though they may look different. Ex: -2x² (4x+5) + 12 = -8ᶟ -10x² + 12

Cubic Polynomial Sample

f(x) = x^3 + 3x² - 2x + 6 f(x) = x²(x + 3) + 2(x + 3) f(x) = (x + 3)(x² + 2) (x + 3)(x² + 2) = 0 For more examples: https://www.mathsisfun.com/algebra/polynomials-solving.html

Word problems using a variable and a constant

these problems include one or more items that vary and one or more items that remain constant.

Solving single variable equations: balancing

whatever we do to one side of an equation, we must also do to the other side. Ask yourself, "what operation is being done to the variable?" then do the algorithmic opposite. 2a + 3 (a + 2) = 4a + 12 2a + 3a + 6 = 4a + 12 5a + 6 = 4a + 12 5a - (4a) + 6 = 4a - (4a) + 12 a + 6 - (6) = 12 - (6) a = 6 .

Axis Symmetry for Parabola

x = -b/2a Ex: x² + 5x + 6, a = x² b = 5 c = 6 x = -5/2(1) = -5/2 is the x-coordinate. y = (-5/2)² + 5(-5/2) + 6 y = 25/4 + -25/2 + 6, Find LCD = 4. y = 25/4 - 50/4 + 24/4 y = -25/4 + 24/4 y = -1/4 is the y-coordinate. Vertex of the parabola (-5/2, -1/4)

Quadratic Formula

x = −b ± √(b2 − 4ac)/2a "A negative boy was thinking yes or no about going to a party, at the party he talked to a square boy but not to the 4 awesome chicks. It was all over at 2 am." This formula is used to solve for the roots or zeros of quadratic equations.

Factoring Polynomials example 2

x² + 7x + 6 = (x + 6)(x + 1) because 6 x 1 = 6 and 6 + 1 adds to 7.

Find the linear equation shown on the coordinate graph

y = mx + b; m = slope or rise/run or y2 - y1/x2 - x1 and b = y-intercept. (-1,1) and (0,3) 3-1/1-0 = 2/1 = 2 is the slope. y-intercept = 3. y = 2x + 3

Graphing a parabola: simplest equation

y = x²

Negative Function

{(1,2) (1,4) (2,-3)} The domain is associated with 2 or more ranges. That is not a function. However, it is still a relation. But you don't know which member to use as an output.

Positive Function

{(1,2) (2,2) (3,-7)} Each domain is associated with one range.