Studying for College Math 101

Parallel lines always/never have the same slope

Parallel lines always have the same slope

Two points on a line are (2,2) and (3,1). Find the y-intercept of the line.

The y intercept is (0,4)

In the equation y = mx + b, m stands for _____ and b stands for _____

m = slope and b = y-intercept

The line y = -x + 4 will/will not enter the third quadrant

The line y = -x + 4 will not enter the third quadrant

The equation for the line shown below is _____

y = 3x+2

If a line is completely horizontal, that line has a(n) undefined/zero slope.

If a line is completely horizontal, that line has a zero slope

Find the slope and y-intercept of the following equation: 2y = 4x + 5

m = 2
y-intercept = (0,5/2)

The slope of the line shown below is _____

The slope of the line is 3 (m=3)

The line y = 3/4x + 5 will be parallel/perpendicular to the line y = -4/3x -3

These lines will be perpendicular

Two points on a line are (2,2) and (3,1). Find the slope of the line.

The slope of the line is -1 (m = -1)

Solve this system of equations
2x + y = 4
x + 2y = 2

(2,0)

The x and y intercepts of 4x + 2y = 8 are _____

x-intercept = (2,0) and y-intercept = (0,4)

If a line is completely vertical, that line has a(n) undefined/zero slope.

If a line is completely vertical, that line has an undefined slope

The three main ways to solve a system of equations are _____, _____ , and _____ .

Graphing, substitution, and elimination are the three main ways to solve a system of equations

The method to evaluate an absolute value expression.

Evaluate as usual. Treat the Absolute Value symbols as parentheses and use PEMDAS.

x|2+y|- z when x=2, y=-5 and z=3

(2)|2+ (-5)| - 3

(2) |2-5| -3

2|-3| -3

2(3) -3 {absolute value of -3 is 3}

6-3 = 3

The process of finding a determinant with matrices larger than 2x2.

Calculate multiple smaller determinants first and combine them by alternating subtraction and addition of the determinants.

The process of multiplying matrices.

M=matrix, R=row, C=column, A=answer

M1R1 x M2C1; Add products = 1st entry R1A

M1R1 x M2C2; Add products = 2nd entry R1A

M1R2 x M2C1; Add products = 1st entry R2A

continue

The part of an absolute value equation that tells you the graph will shift vertically.

When adding or subtracting outside the absolute value sign:

y = |x| + 3 (shifts up 3 places)

The importance of Square Matrices.

You can only calculate a Determinant on a square matrix.

Absolute Value

The magnitude of a number (distance from 0) regardless of sign.

Add:

|3 4| + |-2 5|

3+-2 = 1

4 + 5 = 9

thus;

|1 9|

Transformation (in absolute value graphs)

Anything that changes the way an absolute value graph looks.

Translation (in graphing)

Operations that move the graph of an absolute value equation horizontally or vertically.

The three things that will occur in the graph of this equation.

y=-4|x-3|-3

Reflected due to the absolute value being multiplied by -4.

Translated right 3 places due to the -3 in the absolute value.

Translated down 3 places due to the -3 outside the absolute value.

Find the solution to -t/8 > 3

The solution is t < -24

Write the inequality shown below.

The inequality is y > -x/4 - 2.

Write the inequality shown below.

The inequality in slope-intercept form is: y <- 3x + 7

Find the solution to 2n - 9 > 1

The solution is n > 5

Find the solution.

|x + 5| < 9

The solution is all x-values such that x > -14 and x < 4.

Find the solution to the following inequality.

6x - 3 < - 15

The solution is: x < - 3

Find the solution to 4 - 3x < 10

The solution is x > -2

How do you graph the inequality y > x - 4?

Find the solution to -4x/5 < 4

The solution is x > -5

Represent the following in interval notation.

x > -3 and x < 5

The interval notation is (-3, 5).

Find the solution.

|x - 9| > - 7

The solution is: x < - 2 or x > - 16

A dashed line is used to graph any two-variable inequality that contains either of these symbols.

When a two-variable inequality contains either less than (<) or greater than (>), a dashed line will be used to graph the boundary line.

Find the solution to x + 4 < 12

The solution is x < 8

Solve using complete the square:

x² + 2 = 6x - 2

1. Standard form: x² - 6x + 4 = 0

2. (b/2)² = (-6/2)² = 9

3. to make c = 9 add 5 to both sides

4. x² - 6x + 9 = 5

5. factor perfect square and solve (x - 3)(x - 3) = 5

x - 3 = 5

x = 8

Process to factor quadratics, using x2 + 7x + 6 as an example.

Factor constant term (6): 6*1, 2*3, -6*-1, -2*-3

Add factors to find middle term number (7x): 6 + 1 = 7

Separate factors into product of binomials

(x + 6)(x + 1)

Process of completing the square

Change the c (using algebraic rules) so that it equals (b/2)2.

Use the quadratic formula to determine the discriminant of: x2 - 3x +1

Discriminant of Quadratic Formula is b² - 4ac

-3² - 4(1)(1) = 5

Factor: x2 - 4

constant factors: 2 * -2

sum of factors=0=middle term of (0x)

use factors in binomials

(x + 2)(x - 2)

Maximum of a Parabola

Highest point on the U shaped graph.

Quadratic Formula

The quadratic formula:

x = -b+- v-b2 - 4ac/2a

Find the solutions:

x² - 3x + 2 = 0

Factor into binomials:

(x - 2)(x - 1) = 0

x - 2 = 0

x - 1 = 0

x = 2 and x = 1

Perfect Square Binomial

A binomial that is squared: (x - h)2.

The graph of which of these equations are parabolas:

a. y = x + 1

b. y = x⁵ - 1

c. y = x² - 6

d. y = 2x²

Quadratics with squared terms only lead to parabolas, so c. and d. are parabolas.

Leading Coefficient

The coefficient (number being multiplied) of the first term in a polynomial.

Product

Answer to a multiplication problem.

Factor: 3x2 - 14x - 5

+ = -14

* = (3 * -5) = -15

numbers are 1 and -15

factor -15 (3 * -5)

final: (3x + 1)(x - 5)

Type of equation that leads to a parabola graph.

Quadratics with squared terms such as x2.

Purpose of completing the square

To change a standard form equation into a vertex form equation.

Identify the parts of the graph in this vertex form equation.

y= -5(x + 1)² - 2

-5 = concave down

y-intercept = -7 (substitute x = 0)

axis of symmetry x = - 1 (from equation)

Discriminant of the Quadratic Formula

Inside the sqrt sign: b2 - 4ac

FOIL

A binomial (expression with two terms) is multiplied by a binomial.

Multiply: First terms in each binomial, Outside terms, Inside terms, then Last terms.

Axis of Symmetry

Line at which a graph could be folded and it would be the same on both sides. It goes straight down the middle.

Process to solve quadratics that are not in standard form.

Use algebraic rules to rewrite the equation in proper standard form, then solve with the zero product property or the quadratic formula.

Solve:

(x+2)² = 8 + 4x

Expand and solve for standard form:

x² + 4x + 4 - 8 - 4x = 0

x² - 4 = 0

Solve with zero product

(x - 2)(x + 2) = 0

x = 2 or -2

Zero Product Property

Solving quadratics by factoring into a product of binomials equal to zero. This offers two binomials to set equal to zero in order to get solutions to the original quadratic.

Give the roots of this function:

f(x) = x² - 4x + 4

The roots are both 2.

(x - 2)(x - 2) = 0

x - 2 = 0

x = 2

The process of factoring non-1 leading coeffecient trinomials.

Need 2 numbers whose sum is the middle coefficient.

Product of the numbers =product of the constant and the leading coefficient.

Work backwards to get correct coefficients for each binomial.

A trinomial can be factored into a Perfect Square when...

ax² + bx + c

and

(b/2)² = c

The Standard Form of a Quadratic Equation

y = ax2 + bx + c

Root of a Function

A solution when the function equals 0. The roots of a function are the numbers, which when multiplied together give the original function.

Identify the parts of the graph in this standard form equation:

y = ax² + bx + c

y = ax² + bx + c

a = concave up (pos) or down(neg)

y-intercept = c

axis of symmetry: x = -b/2a

vertex: sub x = -b/2a in to get y value for coordinate point (x,y)

Find the vertex of: y = 3x2 + 6x +1

Axis of sym: x = -(6)/2(3) = -1

Sub: y = 3(-1)² + 6(-1) + 1 = -2

Vertex point (-1,-2)

Values that when multiplied together give an original number.

Ex: the factors of 15 are 5 and 3 (5*3=15)

Factors

Characteristics of a Parabola:

• Looks like a U shape

• Has 1 vertex

• Always symmetrical

• Shape of the path of a thrown object

Multiply:

(x - 5)(x + 2)

F: x * x = x²

O: x*2=2x

I: -5 * x = -5x

L: -5 * 2 = -10

x² - 3x - 10

Vertex of a Parabola

The point at which a parabola changes direction.

Concave down it is the maximum.

Concave up it is the minimum.

Cube root is the same as

raising to the power of 1/3

Exponent Property:

Product of Powers

Add

x² * x⁴ = x²⁺⁴

x⁶

Parts of a Rational exponent

x^2/3

Denominator of the fraction is the ROOT

Numerator is the exponent used on the variable inside the root symbol

Cube root of x²

Divide:

(x³ - 1) divided by (x² + 1)

Step 1: x² + 0x + 1 into x³ + 0x² + 0x - 1

Step 2: x³/x² = x (first ans. term)

Step 3: x(x² + 0x + 1) = x³ + 0x² + x

Step 4: sub. & bring down= -x - 1

Step 5: x² doesn't go into -x

Ans: x + (-x - 1)/(x² + 1)

Simplify:

3×5y5/6×2y7

Break it up to simplify each part.

3/6 = 1/2

x⁵/x² = x³

y⁵/y⁷ = 1/y²

answer: x³/2y²

Simplify a math expression

Reduce to lowest terms

Adding Polynomials

Collect like terms

Five Main Exponent Properties

Product of Powers

Power of Powers

Quotient of Powers

Power of a Product

Power of a Quotient

Convert to Rational exponents:

fourth root of the square of 3x

(3x)2/4

Exponent Property:

Power of a Quotient

Distribute the exponent:

(x/y)³ = x³/y³

Subtracting Polynomials

Change the sign of the terms in the second (subtracted) polynomial and collect like terms.

Synthetic division works for

dividing by first degree binomial only

Anything raised to the power of zero

Equals 1.

Exponent Properties:

Quotient of a Power

Subtract:

x⁵ / x³ = x^5-3

x²

Simplify:

3(p2/3) 2 (q1/2) 2/3/3p

Use exponent rules to simplify: 3p4/3q2/6/3p = 3/3 x p4/3/p x q2/6/1 = p1/3q2/6

Rational Number

Numbers written as a fraction

Simplify:

9^2/3

cube root of 9 squared=

cube root of 81= 4.33

Use synthetic division:

x² + 4 / x - 1

1 | 1 0 4 (opposite of constant in the divisor divided into coefficients in order in the dividend)

bring down 1.

1 * 1 = 1, 0 + 1 = 1

1 * 1 = 1, 4 + 1 = 5

ans 1 1 5

dividend is 2nd degree; ans is 1st degree

x + 1 R5

Add:

(2x³ + 3x² -1) + (x² - 1)

2x³ + 3x² + x² - 1 - 1

2x³ + 4x² - 2

Synthetic division

method of dividing polynomials by using only the coefficients of the terms

Deal with negative exponents

By flipping them to a positive:

x^-2 = 1/x²

1/x^-2 = x²

Steps to divide polynomials with long division

Same as any other long division:

Divide the first terms.

Multiply the result from the divisor, then subtract it from the dividend.

Repeat!

Matching functions to graphs of cubics, quartics, quintics and beyond

Step 1: substitute 0 for x to find y-intercept

Step 2: find values for x = 1 and -1

Step 3: plot more points (opposite matches is great 2 and -2 etc) until sketch becomes clear

Exponent Property:

Power of a Product

Distribute the exponent:

(xy)²

x²y²

Exponent Property:

Power to Power

Multiply

(x²)⁵ = x^2*5

x¹⁰

Subtract:

(x³ - x² + 1) - (x³ - x² + x)

step 1: x³ - x² + 1 - x³ + x² - x

step 2: 1 - x (add like terms and simplify)

Function

A relation in which each input value gives only one possible output value. The graph of a function passes the Vertical Line Test.

For the function g(x) =x2 - 3, find g(4).

g(4) = 13

Using f(x) = 3x and g(x) = x + 4, find the composite function f(g(x)).

The composite function is:

f(g(x)) = 3(x + 4) = 3x + 12

Vertical Line Test

The Vertical Line Test is a way to test whether the points on a graph represent a function. If any vertical line passes through more than one point of the graph, the relation graphed is not a function.