Quadratic Expressions, Equations, and Functions

SAT Math Overview. Topic: 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:

Examples of quadratic expressions: negative 2x squared, 3y squared, 24x squared + 2x, 3x squared + 4xy+y squared.

Be sure you know how to translate a verbal expression into a quadratic expression like the ones shown above. Here are some examples:

“A number times itself.”

x squared

“A number multiplied by one more than the number.”

x times the quantity x + 1 = x-squared + x.

“The product of a number and one less than the number.”

x times the quantity x minus 1 = x squared minus x.

 

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 non-quadratic expression. Here are some examples:

 

Adding and subtracting quadratic expressions results in another quadratic expressions. Multiplying and dividing quadratic expressions results in non-quadratic expressions.

 

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:

The binomial expansion of multiplying the binomials x plus a, and x plus b.

Binomial Expansion

The expansion of a binomial squared.

Binomial Squared

Writing an expression that is difference of squares as the product of two binomials.

Difference of Squares

 

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.

An example of factoring a constant.

Factoring a constant.

An example of factoring a linear term.

Factoring a linear term

An example of factoring a quadratic into a binomial squared.

Factoring a quadratic into a binomial squared

An example of factoring a quadratic into the product of two binomials.

Factoring a quadratic into the product of two linear expressions.

 

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

 

SAT Skill: Working with Quadratic Expressions

 

Example 1

Simplify the following expression.

Simplifying a quadratic expression

Combine like terms and simplify. Keep track of how the subtraction symbol will change the signs of the terms in the expression on the right.

Simplifying a quadratic expression

Example 2

Simplify the following expression.

Simplifying a quadratic expression

Expand the term on the left (use a calculator for the squared decimal terms):

Simplifying a quadratic expression

Combine like terms and simplify. Keep track of how the subtraction sign changes the signs of the terms on the right.

Simplifying a quadratic expression

Example 3

In the equation below, what is a possible value for k?

Factoring a difference of squares.

You should recognize that the term on the left is a difference of squares, which can be rewritten this way:

 

Factoring a difference of squares.

In factored form, you can find the equivalent values for k and a:

k equals 3a equals 2

Example 4

What quadratic expression is equivalent to this expression?

Factoring a linear term.

Factor the numerator, which is a difference of squares, then simplify:

Factoring a linear term.

Example 5

What is this expression in expanded form?

A binomial squared.

Use the binomial squared identity:

An expansion of a binomial squared.

Example 6

Look at the following equation. If a + b = 10, what are two possible values for c?

An example of a binomial expansion.

Expand the term on the left to a quadratic in standard form. 

An example of a binomial expansion.

Compare the two terms in standard form:

Comparing equivalent binomial expansions.

From this we can conclude the following:

Comparing equivalent terms.

We also know that a + b = 10. We can substitute to create this equation:

Solving a quadratic equation.

Factor this quadratic to find the values for b, and corresponding values for a:

Solving a quadratic equation.

Now we can find the possible values for c:

Solving a quadratic equation.

 

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 one-to-one 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.

An image of a function machine.

 

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. 

 

Quadratic Function in Standard Form. A polynomial of degree 2 written in the form shown below, for real numbers a, b, and c. y = a x-squared + bx + c

 

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. 


 

Quadratic Function in Vertex Form. A quadratic function written so as to indicate the coordinates of the parabola's vertex.

 

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.


 

Quadratic Function in Factored Form. A quadratic function written so as to indicate the root or roots of the quadratic. Not all quadratics can be written this way.

 

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. 

Parabola. The graph of a quadratic function is a parabola. The value of a determines its orientation.

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:

Equation of a circle

The coordinates of the center of the circle are (h, k) and the radius of the circle is r.

 

SAT Skill: Working with Quadratic Functions

 

Example 1

In the equation below, a is a positive constant. Write the equation in factored form and write the coordinates of the vertex.

y equals x squared minus 4 a

A parabola in vertex form has this type of equation:

y equals open parentheses x minus h close parentheses squared plus k

Write the given equation in vertex form to find the coordinates of the vertex:

A quadratic function in vertex form.

Next, this equation can be written in factored form as the difference of squares (note the use of the square root sign for the non-squared term):

A quadratic function in factored form.

Example 2

For the function f defined below, a is a constant and f(2) = 60. What is the value of f(5)?

F of x = ax squared + 48

Use the output of f(2) to solve for a:

Evaluating a quadratic for x = 2.

This means that f(x) can be written this way:

F of x equals 3 x squared plus 48

We can now evaluate this function for x = 5: 

Evaluating a quadratic for x = 5.

Example 3

What is the equivalent equation of the parabola that shows the coordinates of point A?

The graph of the quadratic function y = x squared minus 4x plus 8.

The equation of the parabola shown is in standard form. Since point A of the parabola is the vertex, then the equation to write is the vertex form. Use the coordinates (2, 4) to define the vertex form:

y equals open parentheses x minus 2 close parentheses squared plus 4

Example 4

What is the equation of a circle whose center is at (4, 5) and one of the radius endpoints is (7, 9)?

The equation of the circle is of this form:

Writing the equation of the circle with center (4, 5) and one of its radius endpoints at (7,9).

We don’t know the radius of the circle yet, but we have two coordinates for a radius: the center (4, 5) and the endpoint of a radius at (7, 9).

Use the distance formula to find the radius:

Using the distance formula to find the distance between coordinates (4, 5) and (7, 9).

We can now complete the equation of the circle.

open parentheses x minus 4 close parentheses squared plus open parentheses y minus 5 close parentheses squared equals 5 squared

Example 5

A circle has this equation. What is the center of the circle?

x squared minus 2 x plus y squared minus 10 x equals negative 10

When the equation of a circle is written in standard form, the center and radius are easily found:

The equation of a circle with center (h, k) and radius r.

To write the given equation in standard form, use the technique of completing the square. 

Writing the equation of a circle in standard form.

To learn more about the technique of completing the square, click on this link.

 

Quadratic Equations

Quadratic equations are usually written as a quadratic expression in standard form equal to zero.

a x squared plus b x plus c equals 0

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.

a x squared plus b x plus c equals 0

Use the a, b, and c values from the quadratic equation and plug them into the quadratic formula:

Quadratic Formula. Used to solve a quadratic equation. The roots of the quadratic are either real or complex. The values for a, b, and c come from the quadratic function in standard form.

The quadratic formula: x equals fraction numerator negative b plus-or-minus square root of b squared minus 4 a c end root over denominator 2 a end fraction

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.

Discriminant. The part of the quadratic formula used to determine whether a quadratic equation has real roots, and how many.

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:

open parentheses x minus a, close parentheses open parentheses x minus b close parentheses equals 0

The solutions to this equation are x = a and x = b. 

A more simplified version of a factored quadratic can look like this:

x open parentheses x minus a, close parentheses equals 0

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:

open parentheses x minus a, close parentheses squared equals 0

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:

x squared equals 0

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 x-axis twice, then there are two solutions. 

Suppose you are solving this quadratic equation:

 x squared minus 6 x plus 8 equals 0

To find the solution graphically, then graph the corresponding quadratic function.

y equals x squared minus 6 x plus 8

The graph of a parabola that intersects the x-axis at x = 2 and x = 4.

Notice that this parabola intersects the x-axis at x = 2 and x = 4. Those are the solutions to the quadratic equation. In fact, you can rewrite the quadratic in factored form:

open parentheses x minus 2 close parentheses open parentheses x minus 4 close parentheses equals 0

Case 2: One solution. If the graph of the parabola intersects the x-axis once, then there is only one real number solution. 

Suppose you are solving this quadratic equation:

 x squared plus 4 x plus 4 equals 0

To find the solution graphically, then graph the corresponding quadratic function.

y equals x squared plus 4 x plus 4

Graph of a Parabola that intersects the x-axis at x = negative 2.

Notice that this parabola intersects the x-axis at x = -2. This is the solution to the quadratic equation. In fact, you can rewrite the quadratic as a binomial squared:

open parentheses x plus 2 close parentheses squared equals 0

Case 3: No real solutions. If the graph of the parabola doesn’t intersect the x-axis, then there are no real solutions to the quadratic equation. 

Suppose you are solving this quadratic equation:

 x squared minus 4 x plus 6 equals 0

To find the solution graphically, then graph the corresponding quadratic function.

y equals x squared minus 4 x plus 6

Graph of a parabola that doesn't intersect the x-axis, indicating that it has no real roots.

Notice that this parabola doesn’t intersect the x-axis. 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 x-axis, then the parabola has at least one real number solution. These intersection points are also referred to as:

  • x-intercepts
  • 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 worked-out math examples.

 

SAT Skill: Solving Quadratic Equations

Example 1

The parabola with the equation shown below intersects the line with equation y = 16 at two points, A and B. What is the length of segment AB?

y equals open parentheses x minus 8 close parentheses squared

For a question of this type, where references are made to graphs and intersection points, it’s best to draw a diagram to get a better understanding of the problem. The equation shown is of a parabola in vertex form that intersects y = 16. Sketch that.

The graph of the parabola y = the quantity x minus 8 squared and the graph of y = 16. The points A and B are where the graphs intersect.

We know the y coordinates for A and B; in both cases it’s 16. To find the corresponding x-coordinates, solve this equation:

Solving a quadratic equation

The x-coordinate for point A is x = 4 and the for B it’s x = 12. So the distance from A to B is the difference, or 8.

Example 2

In the quadratic equation below, a is a nonzero constant. The vertex of the parabola has coordinates (c, d). Which of the following is equal to d?

y equals a open parentheses x minus 3 close parentheses open parentheses x plus 5 close parentheses

When a quadratic is written in standard form, the vertex has these coordinates:

Showing the relationship between the values a, b, and c in the standard form and h and k in vertex form.

Write the function in standard form. 

Writing a quadratic function in standard form.

Now find the corresponding x and y coordinates with this equation.

Finding the coordinates of the vertex of a parabola when the equation is written in standard form.

Example 3

The parabola whose equation is shown below intersects the graph of y = x at (0, 0) and (a, a). What is the value of a?

y equals 2 x squared minus 5 x

To get a better understanding of this problem, draw a diagram. 

Graph of the parabola y = 2x squared minus 5x and y = x.

To find the value of a, solve the following equation:

Solving a quadratic equation to find the value of parameter a.

 

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