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### 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…]

### 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…]

### How to Graph a Rational Function with Numerator and Denominator of Equal Degrees

After you calculate all the asymptotes and the *x-* and *y-*intercepts for a rational function, you have all the information you need to start graphing the function. Rational functions with equal degrees in [more…]

### How to Vertically Transform Parent Graphs

When you apply a *vertical* *transformation* to a parent graph, you are stretching or shrinking the graph along the *y**-*axis, which changes its height. A number [more…]

### How to Horizontally Transform Parent Graphs

*When you apply a h**orizontal transformation* to a parent graph, you are stretching or shrinking the graph horizontally, along the *x-*axis. A number multiplying a variable inside a function affects the horizontal [more…]

### How to Translate a Function's Graph

When you move a graph horizontally or vertically, this is called a *translation.* In other words, every point on the parent graph is translated left, right, up, or down. Translation always involves either [more…]

### How to Reflect a Function's Graph

*Reflections* take a 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 Graph Functions with More than One Rule: Piece-wise Functions

Functions with more than one rule (called *p**iece-wise functions*) are broken into pieces, depending on the input. Although a piece-wise function has more than one function, each function is defined only [more…]

### How to Graph a Rational Function with Numerator Having the Higher Degree

After you calculate all the asymptotes and the *x-* and *y-*intercepts for a rational function, you have all the information you need to start graphing the function. Rational functions where the numerator [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 [more…]

### How to Adjust the Domain and Range of Combined Functions

When you begin combining functions (like adding a polynomial and a square root, for example), the domain of the new combined function is affected. The same can be said for the range of a combined function [more…]

### How to Graph the Inverse of a Function

If you’re asked to graph the inverse of a function, you can do so by remembering one fact: a function and its inverse are reflected over the line *y* = *x.* [more…]

### How to Invert a Function to Find Its Inverse

If you’re given a function and must find its inverse, first remind yourself that domain and range swap places in the functions. Literally, you exchange [more…]

### How to Solve an Exponential Equation with a Variable on One or Both Sides

Whether an exponential equation contains a variable on one or both sides, the type of equation you’re asked to solve determines the steps you take to solve it. [more…]

### How to Solve an Exponential Equation by Taking the Log of Both Sides

Sometimes you just can’t express both sides of an exponential equation as powers of the same base. When facing that problem, you can make the exponent go away by taking the log of both sides. For example [more…]