Algebra and Functions Flashcards
Logarithm
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
Rules of Logarithms
- loga^b=b(loga)
- loga + logb = log(ab)
- loba – logb = log(a/b)
Linear Equation
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.
Slope-intercept Form
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
Point-Slope Form
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.
Slope Formula
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.
Intercept Form
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.
Horizontal and Vertical Lines
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.
Methods for Solving Simultaneous Linear Equations
- Substitution
- Addition
- Graphing
Substitution Method (Simultaneous Linear Equations)
- Select one of the two equations and rearrange the equation to isolate one of the unknown variables
- Substitute the equation for the isolated variable in the other equation and solve for the other variable
- Plug the known variable into either equation to solve for the unknown variable
Addition Method (Simultaneous Linear Equations)
- Manipulate the equations in a way that allows you to eliminate one of the variables from adding the equations together
- Plug the known variable into either equation to solve for the unknown variable
Graphing Method (Simultaneous Linear Equations)
Graph both equations and find the x and y values where the lines intersect
Dependent Equations (Simultaneous Linear Equations)
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.
General Solution (Simultaneous Linear Equations)
- Re-write either equation to solve for y in terms of x
- Put the solution in the form (x, y), substituting y for the solution for y in terms of x
Example: (x, 2x + 1)
Consistent System of Linear Equations
A system of linear equations is considered consistent if there is only one solution for the system.
Inconsistent System of Linear Equations
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.
Parallel Lines
Parallel lines are two different lines that have the exact same slope, so they never intersect
Perpendicular Lines
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.
Steps to Solving Absolute Value Equations
- Isolate the absolute value expression to one side of the equation
- 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
- Solve both equations to find the solution(s)
Quadratic Equations
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
Methods for Solving Quadratic Equations
1) Factoring
2) Difference of two squares
3) Quadratic formula
Factoring (Quadratic Equations)
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.
Difference of Two Squares (Quadratic Equations)
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
Quadratic Formula
x = [-b +/- ⎷(b2 - 4ac) ] / 2a
Discriminant
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
Relation
A set of points, or ordered pairs
Example: R = {(1,2), (2,4), (0,3), (5,6)}
Ordered Pair
Ordered pairs represent points on a graph and are in the form (x,y)
Domain of a Relation
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}
Range of a Relation
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}
Function
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.
Piecewise Function
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
5 Ways to Represent Domain of a Function
- Inequality notation
- Interval notation
- Set notation
- Graph on a number line
- Written description
Domain of a Function Rules
- If the function contains a fraction, the the domain excludes any values of x that would cause the denominator to be equal to zero
- If the function contains an even root, the domain excludes any values of x that would cause the radicand to be negative
Inverse of a Function
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.
Exponential Functions
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.
Domain and Range of Exponential Function
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.
Growth Curve
An exponential function in which the constant, b, is greater than 1.
Decay Curve
An exponential function in which the constant, b, is fractional value between zero and one.
Composition of Functions
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)
Combining Functions Algebraically
(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)
Graph of Function Transformation - Vertical Shift
y = f(x) + a, moves the graph up a units
y = f(x) - a, moves the graph down a units
Graph of Function Transformation - Horizontal Shift
y = f(x + a), moves the graph a units left
y = f(x - a), moves the graph a units right
Graph of Function Transformation - Vertical Stretch / Compression
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
Graph of Function Transformation - Reflection Across X-axis
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
Graph of Function Transformation - Horizontal Stretch / Compression
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
Graph of Function Transformation - Reflection Across Y-axis
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
Graph of Function Transformation - 90° rotation
90° rotation clockwise: each point (x, y) becomes (y, -x)
90° rotation counterclockwise: each point (x, y) becomes (-y, x)
