Functions: Cheat Sheet

Essential Concepts

  • A relation is a set of ordered pairs. A function is a specific type of relation in which each domain value, or input, leads to exactly one range value, or output.
  • Function notation is a shorthand method for relating the input to the output in the form [latex]y=f\left(x\right)[/latex].
  • In table form, a function can be represented by rows or columns that relate to input and output values.
  • To evaluate a function we determine an output value for a corresponding input value. Algebraic forms of a function can be evaluated by replacing the input variable with a given value.
  • To solve for a specific function value, we determine the input values that yield the specific output value.
  • An algebraic form of a function can be written from an equation.
  • Input and output values of a function can be identified from a table.
  • Relating input values to output values on a graph is another way to evaluate a function.
  • A function is one-to-one if each output value corresponds to only one input value.
  • A graph represents a function if any vertical line drawn on the graph intersects the graph at no more than one point.
  • A graph represents a one-to-one function if any horizontal line drawn on the graph intersects the graph at no more than one point.
  • Fundamental toolkit functions include linear, quadratic, polynomial, exponential, and logarithmic functions, among others. Each type of function has a distinct graph shape. For example, linear functions have straight-line graphs, quadratic functions have parabolic graphs, and exponential functions have curved graphs that increase or decrease rapidly.
  • The ordered pairs given by a linear function represent points on a line.
  • Linear functions can be represented in words, function notation, tabular form and graphical form.
  • The rate of change of a linear function is also known as the slope.
  • An equation in slope-intercept form of a line includes the slope and the initial value of the function.
  • The initial value, or [latex]y[/latex]-intercept, is the output value when the input of a linear function is zero. It is the [latex]y[/latex]-value of the point where the line crosses the [latex]y[/latex]-axis.
  • An increasing linear function results in a graph that slants upward from left to right and has a positive slope. A decreasing linear function results in a graph that slants downward from left to right and has a negative slope. A constant linear function results in a graph that is a horizontal line.
  • Analyzing the slope within the context of a problem indicates whether a linear function is increasing, decreasing, or constant.
  • The slope of a linear function can be calculated by dividing the difference between [latex]y[/latex]-values by the difference in corresponding [latex]x[/latex]-values of any two points on the line.
  • The slope and initial value can be determined given a graph or any two points on the line.
  • Point-slope form is useful for finding the equation of a linear function when given the slope of a line and one point. It is also convenient for finding the equation of a linear function when given two points through which a line passes.
  • The equation for a linear function can be written in slope-intercept form if the slope [latex]m[/latex] and initial value [latex]b[/latex] are known.
  • Linear functions may be graphed by plotting points or by using the y-intercept and slope.
  • Graphs of linear functions may be transformed by shifting the graph up, down, left, or right as well as using stretches, compressions, and reflections.
  • The y-intercept and slope of a line may be used to write the equation of a line. The x-intercept is the point at which the graph of a linear function crosses the x-axis.
  • Horizontal lines are written in the form, [latex]f(x)=b[/latex]. Vertical lines are written in the form, [latex]x=b[/latex].
  • Transformations involve shifting, stretching, or compressing the graph of a function in various ways. For linear functions, these transformations can include vertical shifts, and vertical stretches or compressions.
    • Vertical Shift: Changing the y-intercept [latex]b[/latex] in [latex]f(x) = mx + b[/latex] results in a vertical shift of the line. If [latex]b[/latex] increases, the line shifts up; if [latex]b[/latex] decreases, it shifts down.
    • Vertical Stretch or Compression: Altering the slope [latex]m[/latex] affects the steepness of the line. A larger absolute value of [latex]m[/latex] indicates a steeper slope. A positive [latex]m[/latex] indicates an upward slope, while a negative [latex]m[/latex] indicates a downward slope.
  • A polynomial function of degree two is called a quadratic function.
  • The axis of symmetry is the vertical line passing through the vertex.
  • Quadratic functions are often written in general form. Standard or vertex form is useful to easily identify the vertex of a parabola. Either form can be written from a graph.
  • The vertex can be found from an equation representing a quadratic function.
  • The domain of a quadratic function is all real numbers. The range varies with the function.
    • If the parabola has a minimum, the range is given by [latex]f\left(x\right)\ge k[/latex], or [latex]\left[k,\infty \right)[/latex].
    • If the parabola has a maximum, the range is given by [latex]f\left(x\right)\le k[/latex], or [latex]\left(-\infty ,k\right][/latex].
  • The graph of a quadratic function is a parabola. A parabola is a U-shaped curve that can open either up or down.
    • The parabola opens upward if [latex]a > 0[/latex] and downward if [latex]a < 0[/latex].
  • Transformations of quadratic functions
    • Vertical Shift General Form: Changing the constant [latex]c[/latex] in the equation results in a vertical shift of the parabola. If [latex]c[/latex] increases, the parabola shifts upwards; if [latex]c[/latex] decreases, it shifts downwards.
    • Vertical Shift Standard (Vertex) Form: Changing [latex]k[/latex] results in a vertical shift of the entire parabola. Increasing [latex]k[/latex] shifts the parabola upwards, while decreasing [latex]k[/latex] shifts it downwards.
    • Horizontal Shift: This can be achieved by substituting [latex]x[/latex] with [latex](x - h)[/latex], resulting in a function of the form [latex]f(x) = a(x - h)^2 + k[/latex]. The parabola shifts to the right if [latex]h[/latex] is positive and to the left if [latex]h[/latex] is negative.
    • Stretch/Compression: Changing the value of [latex]a[/latex] causes a vertical stretch or compression. If the absolute value of [latex]a[/latex] is greater than 1, the parabola becomes narrower (stretches), and if it’s between 0 and 1, the parabola becomes wider (compresses).
  • A polynomial function is one whose equation contains only non-negative integer powers on the variable.
  • The polynomial term containing the highest power on the variable is the leading term, and its degree is the number of the power. The leading coefficient of a polynomial is the coefficient of the leading term.
  • The graph of a polynomial function describes a smooth, continuous curve.
  • The domain of all polynomial functions is all real numbers.
  • Even degree polynomial functions describe graphs whose ends both point up or both point down.
  • Odd degree polynomial functions describe graphs whose ends points in opposite directions.
  • The sign of the leading term will determine the direction of the ends of the graph:
    • even degree and positive coefficient: both ends point up
    • even degree and negative coefficient: both ends point down
    • odd degree and positive coefficient: the left-most end points down and the right-most end points up.
    • odd degree and negative coefficient: the left-most end points up and the right-most end points down.
  • Polynomial long division can be used to divide a polynomial by any polynomial with equal or lower degree.
  • The Division Algorithm tells us that a polynomial dividend can be written as the product of the divisor and the quotient added to the remainder.
  • Synthetic division is a shortcut that can be used to divide a polynomial by a binomial of the form [latex]x-k[/latex].
  • To find [latex]f\left(k\right)[/latex], determine the remainder of the polynomial [latex]f\left(x\right)[/latex] when it is divided by [latex]x-k[/latex].
  • k is a zero of [latex]f\left(x\right)[/latex] if and only if [latex]\left(x-k\right)[/latex]  is a factor of [latex]f\left(x\right)[/latex].
  • Each rational zero of a polynomial function with integer coefficients will be equal to a factor of the constant term divided by a factor of the leading coefficient.
  • When the leading coefficient is 1, the possible rational zeros are the factors of the constant term.
  • Synthetic division can be used to find the zeros of a polynomial function.
  • Every polynomial function has at least one complex zero.
  • Allowing for multiplicities, a polynomial function will have the same number of factors as its degree. Each factor will be in the form [latex]\left(x-c\right)[/latex] where c is a complex number.
  • Another way to find the [latex]x[/latex]–intercepts of a polynomial function is to graph the function and identify the points where the graph crosses the [latex]x[/latex]-axis.
  • The multiplicity of a zero determines how the graph behaves at the [latex]x[/latex]-intercept.
    • The graph of a polynomial will cross the [latex]x[/latex]-axis at a zero with odd multiplicity.
    • The graph of a polynomial will touch and bounce off the [latex]x[/latex]-axis at a zero with even multiplicity.
  • The graph of a polynomial function changes direction at its turning points.
  • A polynomial function of degree [latex]n[/latex] has at most [latex]n– 1[/latex] turning points.
  • To graph polynomial functions, find the zeros and their multiplicities, determine the end behavior, and ensure that the final graph has at most [latex]n– 1[/latex] turning points.
  • Graphing a polynomial function helps to estimate local and global extremas.

Glossary

average rate of change

the slope of a line between two points on the graph of a function, calculated via a ratio of the change in function output over the corresponding change in function input

continuous function

a function whose graph can be drawn without lifting the pen from the paper because there are no breaks in the graph

degree of a polynomial

highest power of the variable that occurs in the polynomial

domain

the set of all input values into a function; the set of first components in the ordered pairs and each value in it is an input or independent variable, often labeled [latex]x[/latex]

end behavior

the behavior of the graph of a function as the input decreases without bound and increases without bound

function

a relation in which each possible input value leads to exactly one output value

function notation

a notation used for representing output as a function of input, [latex]y=f(x)[/latex], that is [latex]y[/latex] is a function of [latex]x[/latex]

graph transformation

involves shifting, stretching, or flipping its shape to create a new representation

horizontal line test

a method of testing whether a function is one-to-one by determining whether any horizontal line intersects the graph more than once

input

each object or value in a domain that relates to another object or value by a relationship known as a function

leading coefficient

the coefficient of the leading term

leading term

the term containing the variable with the highest power

linear function

characterized by a constant rate of change and can be represented as a polynomial of degree [latex]1[/latex]

modeling

the process of translating real-world problems into mathematical terms and solving them

multiplicity

the number of times a given factor appears in the factored form of the equation of a polynomial; if a polynomial contains a factor of the form [latex]{\left(x-h\right)}^{p}[/latex], [latex]x=h[/latex] is a zero of multiplicity [latex]p[/latex].

one-to-one function

a function in which each output value corresponds to exactly one input value

output

each object or value in the range that is produced when an input value is entered into a function

range

the set of all output values of a function; the set of second components in the ordered pairs and each value in the range is an output or dependent variable, often labeled [latex]y[/latex]

Rational Zero Theorem relation

a set of ordered pairs

slope

the ratio of the change in output values to the change in input values; a measure of the steepness of a line

synthetic division

a shortcut method that can be used to divide a polynomial by a binomial of the form [latex]x – k[/latex]

term of a polynomial function

[latex]{a}_{i}{x}^{i}[/latex]

vertex

the point at which a parabola changes direction, corresponding to the minimum or maximum value of the quadratic function

vertical line test

determines if a relation is a function by checking that no vertical line intersects the graph more than once

[latex]x[/latex]-intercept

value of [latex]x[/latex] where [latex]f(x) = 0[/latex]

[latex]y[/latex]-intercept

the value of a function when the input value is zero; the point at which the graph crosses the horizontal axis; also known as initial value

zeros

the [latex]x[/latex]-intercepts of a quadratic equation

Key Equations

axis of symmetry

[latex]x=-\dfrac{b}{2a}[/latex], where [latex]a[/latex] and [latex]b[/latex] are coming from the general form of a quadratic function 

calculating slope

[latex]m=\dfrac{\text{change in output (rise)}}{\text{change in input (run)}}=\dfrac{\Delta y}{\Delta x}=\dfrac{{y}_{2}-{y}_{1}}{{x}_{2}-{x}_{1}} \Rightarrow \dfrac{f(x_2)-f(x_1)}{x_2 - x_1}[/latex]

Division Algorithm

given a polynomial dividend [latex]f\left(x\right)[/latex] and a non-zero polynomial divisor [latex]d\left(x\right)[/latex] where the degree of [latex]d\left(x\right)[/latex] is less than or equal to the degree of [latex]f\left(x\right)[/latex], there exist unique polynomials [latex]q\left(x\right)[/latex] and [latex]r\left(x\right)[/latex] such that [latex]f\left(x\right)=d\left(x\right)q\left(x\right)+r\left(x\right)[/latex] where [latex]q\left(x\right)[/latex] is the quotient and [latex]r\left(x\right)[/latex] is the remainder. The remainder is either equal to zero or has degree strictly less than [latex]d\left(x\right)[/latex].

Factor Theorem

[latex]k[/latex] is a zero of polynomial function [latex]f\left(x\right)[/latex] if and only if [latex]\left(x-k\right)[/latex]  is a factor of [latex]f\left(x\right)[/latex]

general form of a polynomial function

[latex]f\left(x\right)={a}_{n}{x}^{n}+\dots+{a}_{2}{x}^{2}+{a}_{1}x+{a}_{0}[/latex]

general form of a quadratic function

[latex]f\left(x\right)=a{x}^{2}+bx+c[/latex]

point-slope form

[latex]y-{y}_{1}=m\left(x-{x}_{1}\right)[/latex]

Remainder Theorem

if a polynomial [latex]f\left(x\right)[/latex] is divided by [latex]x-k[/latex] , then the remainder is equal to the value [latex]f\left(k\right)[/latex]

slope-intercept form

[latex]\begin{array}{lll}\text{Equation form}\hfill & y=mx+b\hfill \\ \text{Function notation}\hfill & f\left(x\right)=mx+b\hfill \end{array}[/latex]

standard form (vertex form) of a quadratic function

[latex]f\left(x\right)=a{\left(x-h\right)}^{2}+k[/latex]

the [latex]x[/latex]-coordinate of the vertex

[latex]h=-\dfrac{b}{2a}[/latex], where [latex]a[/latex] and [latex]b[/latex] are coming from the general form of a quadratic function 

the [latex]y[/latex]-coordinate of the vertex

[latex]k=f\left(h\right)=f\left(-\dfrac{b}{2a}\right)[/latex], where [latex]a[/latex] and [latex]b[/latex] are coming from the general form of a quadratic function