Calculus

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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 piece-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 Determine the Length of an Arc

Knowing how to calculate the circumference of a circle and, in turn, the length of an arc — a portion of the circumference — is important in pre-calculus because you can use that information to analyze [more…]

How to Graph a Cosine Function

The parent graph of cosine looks very similar to the sine function parent graph, but it has its own sparkling personality (like fraternal twins). Cosine graphs follow the same basic pattern and have the [more…]

How to Calculate an Angle Using Reciprocal Trigonometric Functions

Three trigonometric ratios — secant, cosecant, and cotangent — are called reciprocal functions because they're the reciprocals of sine, cosine, and tangent. These three functions open up three more ways [more…]

How to Calculate an Angle Using Inverse Trigonometric Functions

Almost every function has an inverse. An inverse functionbasically undoes a function. The trigonometric functions sine, cosine, and tangent all have inverses, and they're often called [more…]

How to Draw Uncommon Angles

Many times in your journey through trigonometry — actually, all the time — drawing a figure will help you solve a given problem. So what do you do if you're asked to draw an angle that has a measure greater [more…]

How to Work with 30-60-90-Degree Triangles

All 30-60-90-degree triangles have sides with the same basic ratio. If you look at the 30–60–90-degree triangle in radians, it translates to the following: [more…]

How to Graph Polynomials

Although it may seem daunting, graphing polynomials is a pretty straightforward process. Once you have found the zeros for a polynomial, you can follow a few simple steps to graph it. [more…]

How to Find a Greatest Common Factor in a Polynomial

No matter how many terms a polynomial has, you always want to check for a greatest common factor (GCF) first. If the polynomial has a GCF, factoring the rest of the polynomial is much easier because once [more…]

How to Use the FOIL Method to Factor a Trinomial

For polynomials with a nonprime leading coefficient and constant term, you can use a procedure called the FOIL method of factoring (sometimes called the [more…]

How to Factor a Perfect Square

FOIL stands for multiply the first, outside, inside, and last terms together. When you FOIL a binomial times itself, the product is called a perfect square. [more…]

How to Factor a Difference of Squares

When you FOIL (multiply the first, outside, inside, and last terms together) a binomial and its conjugate, the product is called a difference of squares. [more…]

How to Break Down a Cubic Difference or Sum

After you’ve checked to see if there’s a Greatest Common Factor (GCF) in a given polynomial and discovered it’s a binomial that isn’t a difference of squares, you should consider that it may be a difference [more…]

Factoring Four or More Terms by Grouping

When a polynomial has four or more terms, the easiest way to factor it is to use grouping. In this method, you look at only two terms at a time to see if any techniques become apparent. For example, you [more…]

Finding the Roots of a Factored Equation

In pre-calculus, you can use the zero-product property to find the roots of a factored equation. After you factor a polynomial into its different pieces, you can set each piece equal to zero to solve for [more…]

How to Solve a Quadratic Equation When It Won’t Factor

When asked to solve a quadratic equation that you just can’t seem to factor (or that just doesn’t factor), you have to employ other ways of solving the equation, such as by using the quadratic formula. [more…]

How to Complete the Square

Completing the square comes in handy when you’re asked to solve an unfactorable quadratic equation and when you need to graph conic sections (circles, ellipses, parabolas, and hyperbolas). [more…]

How to Find the Real Roots of a Polynomial Using Descartes’s Rule of Signs

If you know how many total roots a polynomial has, you can use a pretty cool theorem called Descartes’s rule of signsto count how many roots are real numbers [more…]

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