Quadratic Expressions, Equations, and Functions
Overview
Quadratic Expressions, Equations, and Functions are found throughout the SAT, so it’s very important to be comfortable with all the key aspects of this topic. In particular, make sure you know these concepts:
 Quadratic Expressions
 Quadratic Functions
 Standard Form
 Vertex Form
 Factored Form
 Properties of Parabolas
 Quadratic Equations
 Quadratic Formula
 Graphical Solutions
 Discriminant
 Quadratic Models
Let’s look at each of these in more detail:
Quadratic Expressions
Quadratic expressions include a variable whose exponent is 2, although the product of two linear variables (for example, xy) is also quadratic. Here are some examples of quadratic expressions:
Be sure you know how to translate a verbal expression into a quadratic expression like the ones shown above. Here are some examples:






Expressions aren’t equations but they are an important component of quadratic equations. Also, you can add and subtract quadratic expressions and still have a quadratic expression. Multiplying or dividing quadratic expressions will result in a nonquadratic expression. Here are some examples:
Some of the questions you’ll be asked will involve rewriting quadratic expressions to equivalent forms. To simplify this task familiarize yourself with these quadratic identities:






To learn more about quadratic identities, and identities in general, click on this link.
Another important skill involves factoring quadratic expressions. This is a skill that also comes into play when solving quadratic equations. Here are some examples.








To learn more about factoring quadratics, click on this link. Now look at some SATstyle questions that focus on quadratic expressions.

Quadratic Functions
Before studying what a quadratic function is, make sure you are comfortable with the following concepts, which we will also review:
 What a function is
 Independent variable
 Dependent variable
 Domain
 Range
 Different representations of functions
Brief Review of Functions
What Is a Function? A function is a onetoone mapping of input values (the independent variable) to output values (the dependent variable). Click on this link to see a quick tutorial on what a function is. This slide show goes over the following key points:
 For every input value (x), there is a unique output value, f(x).
 Functions can be represented as equations, tables, and graphs.
 A function machine is a useful visual representation of the input/output nature of functions.
Dependent/Independent Variables. When one variable depends on another, then it is the dependent variable. For example, the faster your speed, the farther you travel. Suppose that speed is represented by the variable s and the distance traveled is represented by the variable d.
Here’s how to describe the relationship between s and d:
The faster the speed, the more distance traveled.
Distance is dependent on speed.
Distance is a function of speed.
d = f(s)
When studying functions, make sure you are comfortable telling the difference between the independent variable and dependent variable. Get comfortable using function notation. To learn more about function notation, click on this link.
Domain and Range. A function shows the relationship between two variables, the independent variable and the dependent variable. The domain is the allowed values for the independent variable. The range is the allowed values for the dependent variable. The domain and range influence what the graph of the function looks like.
For a detailed review of what domain and range are, click on this link to learn more. You’ll see definitions of the terms domain and range, as well as examples of how to find the domain and range for given functions.
Multiple Representations of Functions. We mentioned previously that functions can be represented in different ways. In fact, any function can be represented by an equation, usually f(x) equal to some expression; a table; or a graph. For a detailed review of multiple representations of functions, click on this link, to see a slide show that includes examples of these multiple representations.
Quadratic Functions in Standard Form
The most common form of a quadratic function is the standard form.
The standard form is also the most common form used to solve a quadratic equation.
To see examples of graphing quadratic functions in standard form, click on this link. This slide show also includes a video tutorial.
Quadratic Functions in Vertex Form
Another way that a quadratic function can be written is in vertex form.
To see examples of quadratic functions in vertex form, click on this link. This slide show tutorial walks you through the difference between standard form and vertex form. It also includes examples and two Desmos activities where you can graph these two types of equations.
Quadratic Functions in Factored Form
Another way that a quadratic function can be written is in factored form. From your work with quadratic expressions, you saw some techniques for factoring a quadratic. If you can write a quadratic function as the product of linear terms, then it is much easier to solve the corresponding the quadratic equation.
To see examples of quadratic functions in factored form, click on this link.
Graphs of Quadratic Functions
The graph of a quadratic function is known as a parabola.
To learn more about the properties of parabolas and their graphs, click on this link.
Special Case of Quadratics: Equation of a Circle
The equation of a circle is not a quadratic function, but it is a quadratic relation. This is the equation of a circle:
The coordinates of the center of the circle are (h, k) and the radius of the circle is r.

Quadratic Equations
Quadratic equations are usually written as a quadratic expression in standard form equal to zero.
A quadratic equation can have two, one, or zero real number solutions. There are several ways to solve a quadratic. These are the methods we’ll be looking at:
 The Quadratic Formula
 Factoring
 Graphing the Quadratic Function.
Let’s look at the first method, which will work for any quadratic equation.
The Quadratic Formula
When a quadratic equation is written in standard form, like the one shown below, then you can use the quadratic formula to find the solutions to the equation.
Use the a, b, and c values from the quadratic equation and plug them into the quadratic formula:
To learn more about using the quadratic formula to solve quadratic equations, click on this link. This slide show includes a video overview of the quadratic formula and a number of detailed math examples.
Before using the quadratic formula, calculate the discriminant, which is the term under the square root sign of the quadratic formula.
To see examples of using the discriminant, click on this link.
Factoring
You’ve already seen how to factor quadratic expressions into the product of linear terms. That same idea can be used to factor certain quadratic expressions in order to find the solutions to the equation.
A factored quadratic equation will look something like this:
The solutions to this equation are x = a and x = b.
A more simplified version of a factored quadratic can look like this:
The solutions to this equation are x = 0 and x = a.
The previous two examples both had two solutions. There is a factored form that has one solution:
This is the case of the binomial squared. In this case the solution to the equation is x = a.
The simplest example of the binomial squared is this:
The solution to this is x = 0.
If a quadratic cannot be easily factored, then you should use the quadratic formula or graph the quadratic.
To see examples of using factoring to solve a quadratic equation, click on this link.
Solving by Graphing
A visual approach to solving quadratic equations is to graph the parabola. There are three cases to look at.
Case 1: Two solutions. If the graph of the parabola intersects the xaxis twice, then there are two solutions.
Suppose you are solving this quadratic equation:
To find the solution graphically, then graph the corresponding quadratic function.
Notice that this parabola intersects the xaxis at x = 2 and x = 4. Those are the solutions to the quadratic equation. In fact, you can rewrite the quadratic in factored form:
Case 2: One solution. If the graph of the parabola intersects the xaxis once, then there is only one real number solution.
Suppose you are solving this quadratic equation:
To find the solution graphically, then graph the corresponding quadratic function.
Notice that this parabola intersects the xaxis at x = 2. This is the solution to the quadratic equation. In fact, you can rewrite the quadratic as a binomial squared:
Case 3: No real solutions. If the graph of the parabola doesn’t intersect the xaxis, then there are no real solutions to the quadratic equation.
Suppose you are solving this quadratic equation:
To find the solution graphically, then graph the corresponding quadratic function.
Notice that this parabola doesn’t intersect the xaxis. When this happens, the quadratic equation doesn’t have real number solutions. It does, however, have complex number solutions, which you can find using the quadratic formula.
Summary of Solving by Graphing. When a parabola intersects the xaxis, then the parabola has at least one real number solution. These intersection points are also referred to as:
 Xintercepts
 Zeros of the Quadratic Function
 Roots of the Quadratic Equation
To learn more about solving a quadratic equation graphically, click on the following link. This includes a video tutorial and several workedout math examples.

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