How to Interpret Function Graphs
You’re going to see dozens and dozens of functions in your study of calculus, and the graphs of those functions can visually express such things as inflation, population growth, and radioactive decay. [more…]
How to Horizontally Transform a Function
You can transform any function into a related function by shifting it horizontally or vertically, flipping it over (reflecting it) horizontally or vertically, or stretching or shrinking it horizontally [more…]
How to Vertically Transform a Function
To transform a function vertically, you add a number to or subtract a number from the entire function, or multiply it by a number. To do something to an entire function, say [more…]
How to Work with Lines in Calculus
A line is the simplest function that you can graph on the coordinate plane. (Lines are important in calculus because when you zoom in far enough on a curve, it looks and behaves like a line.) This figure [more…]
How to Recognize Inverse Functions
You can tell that two functions are inverse functions when each one undoes what the other does. When you graph inverse functions, each is a mirror image of the other. Here are some examples of inverse [more…]
How to Translate a Function's Graph
Moving a function's graph horizontally or vertically is called a translation. In other words, you translate every point on the parent graph left, right, up, or down. The kinds of translations fall into [more…]
How to Reflect a Function's Graph
Reflections of a graph take the parent function and provide a mirror image of it over either a horizontal or vertical line. You’ll come across two types of reflections: [more…]
How to Combine Various Transformations
Certain mathematical expressions allow you to combine stretching, shrinking, translating, and reflecting a function all into one graph. An expression that shows all the transformations in one is [more…]
How to Calculate Outputs for Rational Functions
Rational functions are functions where the variable appears in the denominator of a fraction. The mathematical definition of a rational function is a function that can be expressed as the quotient of two [more…]
How to Operate on Functions
You can operate on (sometimes called combining) functions pretty easily. You may have to perform operations on functions — such as addition, subtraction, multiplication, and division — as well as break [more…]
How to Break Down a Composition of Functions
A composition of functions is one function acting upon another. Think of it like putting one function inside of the other — f (g (x)), for instance, means that you plug the entire g [more…]
Even-Odd Identities in Trigonometric Functions
All functions, including trig functions, can be described as being even, odd, or neither. Knowing whether a trig function is even or odd can help you simplify an expression. These even-odd identities are [more…]
How to Solve Compound Functions Where the Inner Function Is ax + b
Some integrals of compound functions f(g(x)) are easy to do quickly in Calculus. These include compound functions for which you know how to integrate the outer function [more…]
Solve Compound Functions Where the Inner Function Is ax
When figuring Calculus problems, some integrals of compound functions f (g(x)) are easy to do quickly. These include compound functions for which you know how to integrate the outer function [more…]
Classifying Differential Equations by Order
The most common classification of differential equations is based on order. The order of a differential equation simply is the order of its highest derivative. You can have first-, second-, and higher-order [more…]
Distinguishing among Linear, Separable, and Exact Differential Equations
You can distinguish among linear, separable, and exact differential equations if you know what to look for. Keep in mind that you may need to reshuffle an equation to identify it. [more…]
Defining Homogeneous and Nonhomogeneous Differential Equations
In order to identify a nonhomogeneous differential equation, you first need to know what a homogeneous differential equation looks like. You also often need to solve one before you can solve the other. [more…]
Using the Method of Undetermined Coefficients
If you need to find particular solutions to nonhomogeneous differential equations, then you can start with the method of undetermined coefficients. Suppose you face the following nonhomogeneous differential [more…]
