Algebra and Functions

Question 1

Logarithm

Answer 1

The power to which a number must be raised to get another number

Logby = x

x is the power to which b must be raised to get y, which can be re-written as b^x=y

If the base (b) of the log is not specified, the understood base is 10

Question 2

Rules of Logarithms

Answer 2

1. loga^b=b(loga)
2. loga + logb = log(ab)
3. loba – logb = log(a/b)

Question 3

Linear Equation

Answer 3

An equation where the highest power of the unknown variable is 1

Graphs as a straight line

Standard form for a linear equation with one unknown variable is ax + b = 0, where a ≠ 0.

Standard form for a linear equation with two unknown variables is ax + by + c = 0, where a ≠ 0 and b ≠ 0.

Question 4

Slope-intercept Form

Answer 4

Slope-intercept form is a form used for linear equations to facilitate graphing the equation.

y = mx + b, where m represents the slope (m = rise/run) and b represents the y-intercept

Question 5

Point-Slope Form

Answer 5

A linear equation that passes through the point (x1, y1) is written as y – y1 = m(x – x1).

Point-slope form allows us to identify the equation for a line when given any two points on the line.

Question 6

Slope Formula

Answer 6

m = (y1 - y2) / (x1 - x2)

The slope formula allows us to identify the slope of a linear equation when given two points on the line.

Question 7

Intercept Form

Answer 7

A linear equation with an x-intercept of a and a y-intercept of b can be written in slope intercept form as (x/a) + (y/b) = 1.

Intercept form allows us to easily identify the equation of a line when given the x-intercept and the y-intercept of the line.

Question 8

Horizontal and Vertical Lines

Answer 8

All horizontal lines are in the form y = constant and all vertical lines are in the form x = constant.

Horizontal lines always have a slope of zero.

Vertical lines are said to have no slope or that the slope is undefined.

Question 9

Methods for Solving Simultaneous Linear Equations

Answer 9

1. Substitution
2. Addition
3. Graphing

Question 10

Substitution Method (Simultaneous Linear Equations)

Answer 10

1. Select one of the two equations and rearrange the equation to isolate one of the unknown variables
2. Substitute the equation for the isolated variable in the other equation and solve for the other variable
3. Plug the known variable into either equation to solve for the unknown variable

Question 11

Addition Method (Simultaneous Linear Equations)

Answer 11

1. Manipulate the equations in a way that allows you to eliminate one of the variables from adding the equations together
2. Plug the known variable into either equation to solve for the unknown variable

Question 12

Graphing Method (Simultaneous Linear Equations)

Answer 12

Graph both equations and find the x and y values where the lines intersect

Question 13

Dependent Equations (Simultaneous Linear Equations)

Answer 13

Equations that represent the same line but are written in different forms.

When looking to find the solution for a system of linear equations made up of dependent equations, there will be an infinite number of solutions for x and y.

Question 14

General Solution (Simultaneous Linear Equations)

Answer 14

1. Re-write either equation to solve for y in terms of x
2. Put the solution in the form (x, y), substituting y for the solution for y in terms of x

Example: (x, 2x + 1)

Question 15

Consistent System of Linear Equations

Answer 15

A system of linear equations is considered consistent if there is only one solution for the system.

Question 16

Inconsistent System of Linear Equations

Answer 16

A system of linear equations is considered inconsistent if it does not have any solutions. A situation where linear equations are inconsistent and do not have any solutions would be parallel lines.

Question 17

Parallel Lines

Answer 17

Parallel lines are two different lines that have the exact same slope, so they never intersect

Question 18

Perpendicular Lines

Answer 18

Two lines that intersect such that the lines form a 90 degree angle.

The slopes of two perpendicular lines are negative reciprocals of each other.

Question 19

Steps to Solving Absolute Value Equations

Answer 19

1. Isolate the absolute value expression to one side of the equation
2. Split the absolute value equation into two equations, one where the solution to the absolute value expression is a positive and the other where it is negative
3. Solve both equations to find the solution(s)

Question 20

Quadratic Equations

Answer 20

Algebraic equations in which the highest power of the unknown variable is two.

Quadratic equations with one unknown variable can be put in the standard form ax^2 + bx + c = 0

Question 21

Methods for Solving Quadratic Equations

Answer 21

1) Factoring
2) Difference of two squares
3) Quadratic formula

Question 22

Factoring (Quadratic Equations)

Answer 22

Factoring can be used to solve quadratic equations when a = 1 and you can find two numbers that multiply to equal c and add together to equal b.

Question 23

Difference of Two Squares (Quadratic Equations)

Answer 23

The difference of two squares method of solving quadratic equations can be used when you have a quadratic equation that consists only of the difference of two perfect squares. When you have the difference of two squares, the two factors are 1) the sum of the square roots and 2) the difference of the square roots

i.e. for x^2 - 4 = 0, the factors are (x+2) and (x-2), so the roots are x = 2 and x = -2

Question 24

Quadratic Formula

Answer 24

x = [-b +/- ⎷(b2 - 4ac) ] / 2a

Question 25

Discriminant

Answer 25

The b^2 - 4ac piece of the quadratic formula. The nature of the discriminant reveals the nature of the roots of a quadratic equation, as described below:

• If the discriminant > 0, the roots are real and unequal
• If the discriminant = 0, the roots are real and equal
• If the discriminant < 0, the roots are complex numbers

If the value of the discriminant is not a perfect square, then the roots of the discriminant are irrational

Question 26

Relation

Answer 26

A set of points, or ordered pairs

Example: R = {(1,2), (2,4), (0,3), (5,6)}

Question 27

Ordered Pair

Answer 27

Ordered pairs represent points on a graph and are in the form (x,y)

Question 28

Domain of a Relation

Answer 28

The list of all x values in a relation

Example: for the relation R = {(1,2), (2,4), (0,3), (5,6)}, domain = {0, 1, 2, 5}

Question 29

Range of a Relation

Answer 29

The list of all y values in a relation

Example: for the relation R = {(1,2), (2,4), (0,3), (5,6)}, range = {2, 3, 4, 6}

Question 30

Function

Answer 30

A relation that for each value of x there is exactly one value for y

Functions can be represented as a written relation, as a graph, or using functional notation.

Question 31

Piecewise Function

Answer 31

A function that is split into more than one formula to use depending on the value of x

Example: f(x) = {x -7 if x < 3, x^2 if x >= 3

Question 32

5 Ways to Represent Domain of a Function

Answer 32

1. Inequality notation
2. Interval notation
3. Set notation
4. Graph on a number line
5. Written description

Question 33

Domain of a Function Rules

Answer 33

1. If the function contains a fraction, the the domain excludes any values of x that would cause the denominator to be equal to zero
2. If the function contains an even root, the domain excludes any values of x that would cause the radicand to be negative

Question 34

Inverse of a Function

Answer 34

The inverse of a function is a function in which the x and y values of the original function are switched. The x values of the original function become the y values of the inverse function, and the y values of the original function are the x values of the inverse function.

The inverse of f(x) is written as f-1(x) (read as “f inverse of x”)

The inverse of a function is not always a valid function. The inverse is only a function if there is only one output value (y value) for each input value (x value), i.e. no repeating x values with different y values.

Question 35

Exponential Functions

Answer 35

The basic form of an exponential function is y = ab^x, b=/ 1
• a is any non-zero real number
• b is any positive real number other than 1

Note: The basic form of an exponential function requires that the base is a constant and the exponent contains the independent variable, x. A function that has the independent variable x raised to a power that is a constant is NOT an exponential function.

Question 36

Domain and Range of Exponential Function

Answer 36

The domain of an exponential function is always going to include all real numbers. If a is positive, the range will be all positive real numbers. If a is negative, the range will be all negative real numbers.

Question 37

Growth Curve

Answer 37

An exponential function in which the constant, b, is greater than 1.

Question 38

Decay Curve

Answer 38

An exponential function in which the constant, b, is fractional value between zero and one.

Question 39

Composition of Functions

Answer 39

A composition of functions occurs when the variable of one function is represented with another function.

If you are looking to define f(g(x)), you substitute the equation for g(x) for x in the f(x) function. If you are looking to find g(f(x)), you substitute the equation for f(x) for x in the g(x) function.

f(g(x)) = (f o g)(x) and g(f(x)) = (g o f)(x)

Question 40

Combining Functions Algebraically

Answer 40

(f + g)(x) = f(x) + g(x)
(f - g)(x) = f(x) - g(x)
(fg)(x) = f(x)g(x)
(f/g)(x) = f(x) / g(x)

Question 41

Graph of Function Transformation - Vertical Shift

Answer 41

y = f(x) + a, moves the graph up a units
y = f(x) - a, moves the graph down a units

Question 42

Graph of Function Transformation - Horizontal Shift

Answer 42

y = f(x + a), moves the graph a units left
y = f(x - a), moves the graph a units right

Question 43

Graph of Function Transformation - Vertical Stretch / Compression

Answer 43

Vertical stretch: y = a(f(x)), where a > 1
Vertical compression: y = a(f(x)), where 0 < a < 1

For graphs that are symmetrical across the y-axis, vertical stretch has the same effect as a horizontal compression and vertical compression has the same effect as a horizontal stretch

Question 44

Graph of Function Transformation - Reflection Across X-axis

Answer 44

y = a(f(x)), where a = -1

If -1 < a < 0, we would have a vertically compressed graph reflected across the x-axis

If a < -1, we would have a vertically stretched graph reflected across the x-axis

Question 45

Graph of Function Transformation - Horizontal Stretch / Compression

Answer 45

Horizontal stretch: y = f(ax), where 0 < a < 1
Horizontal compression: y = f(ax), where a > 1

For graphs that are symmetrical across the y-axis, horizontal stretch has the same effect as a vertical compression, and horizontal compression has the same effect as a vertical stretch

Question 46

Graph of Function Transformation - Reflection Across Y-axis

Answer 46

y = f(ax), where a = -1

For functions that are symmetrical on either side of the y-axis (like y = x^2 and y = |x|), the graph of the function reflected across the y-axis is identical to the original function

If -1 < a < 0, we would have a horizontally stretched graph reflected across the y-axis

If a < -1, we would have a horizontally compressed graph reflected across the y-axis

Question 47

Graph of Function Transformation - 90° rotation

Answer 47

90° rotation clockwise: each point (x, y) becomes (y, -x)
90° rotation counterclockwise: each point (x, y) becomes (-y, x)