# Types of (Math) Relationships, Part 3

The last two SAT Math blog posts have looked at linear and exponential functions. This month we will be focusing on the last common relationship found on the new SAT, quadratic functions. How can you identify quadratic functions? If the equation’s highest exponent is a 2, this is a quadratic function. The general equation of a quadratic function is *f(x) *= *ax^2 + bx + c.* A quadratic equation can take multiple forms, as I will discuss later in the post. Remember, exponential and linear functions appear similarly on the test. Quadratic functions are very different from the other two. Questions involving these functions often focus on either the vertex or the x-intercepts of the equation. These functions can also appear outside of a word problem context. In these questions, the goal will be to solve for the variable.

__Identifying Vertex Questions__

**Example 1**

A store manager estimates that if a video game is sold at a price of *p* dollars, the store will have weekly revenue, in dollars, of from the sale of the video game. Which of the following equivalent forms of *r(p)* shows, as constants or coefficients, the maximum possible weekly revenue and the price that results in the maximum revenue?

Anytime a quadratic question asks about a maximum or a minimum, this will occur at the function’s vertex. This question is simply asking which function is in vertex form.

The general equation of vertex form is *f(x) *= *a(x-h)^2 *+ *k* , with (*h,k*) being the vertex of the equation.

Knowing this, there is only one equation that matches this form of the equation; the answer to this question is D.

If the question were, “What is the maximum weekly revenue?,” you would need to provide the y-coordinate of the vertex. The answer would be $2,500. If the question were, “What is the price that results in the maximum revenue?,” you would need to provide the x-coordinate of the vertex. The answer would be $25.

__Identifying X-Intercept Questions__

X-intercept questions come in more varieties. Sometimes the question will be very straightforward and ask for the x-intercepts, but the question could also ask to solve when y = 0. No matter how the question is asked, the equation needs to be put in factored form to find the answer.

The general equation of factored form is *f(x) *= *a(x-b)(x-c)*, with the x-intercepts being x=b and x=c.

**Example 2**

The graph of which of the following functions in the xy-plane has x-intercepts at -4 and 5?

We can immediately eliminate answers C and D as they are in vertex form, not factored form. When equations are in factored form, the answers are opposite in sign from how they are written. This means that A) has x-intercepts -4 and 5 while B) has x-intercepts 4 and -5.

Therefore, the answer to this question is A.

__Solving Quadratic Equations__

When solving a quadratic equation, the first step should always be to set the equation equal to 0. Once you have achieved this, you will need to factor or use the quadratic formula. In case you have forgotten, the quadratic formula is:

based on the standard quadratic equation: *f(x) *= *ax^2 + bx + c* .

This can work every single time. Sometimes there is tricky algebra involved, but don’t let that intimidate you.

**Example 3**

What are the solutions *x* of *x^2 *– 3 = *x* ?

First, we need to set the equation equal to 0. The easiest way to do this is to subtract *x* from both sides.

*x^2 *– 3 = *x*

*x^2 *– *x *– 3 = 0

Now that the equation is set equal to 0, we must decide if we should factor or use the quadratic formula. Based on the answer choices, this appears to be a quadratic formula problem. For this problem, a=1, b=-1, and c=-3. Plug these values into the formula: . This simplifies to . With final simplification, the answer to this question is D.

__Test Yourself__

**Free Response**

What is the sum of the solution of ?

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