**Graphing Quadratic Functions | Mathematics **

A quadratic function is in the form of** f(x) = ax^{2} + bx + c.** The squared exponent (

*x*

^{2}) makes it quadratic. The variables

*a*,

*b*, and

*c*represent numbers, where

*a*cannot be equal to zero. When a quadratic function is graphed, the shape is a positive (smile shape) or negative (rainbow shape) parabola.

Now, you may be thinking, “You want me to graph *what*?” You’ll find that it’s not as complex as it looks. To prepare for the TASC Test Assessing Secondary Completion™ Mathematics subtest, follow these math tips:

**Find the vertex.**The vertex is found by identifying the coordinates (h, k).**Understand**that*h*= -*b*/2*a*and*k*=*f*(*h*).**Rearrange***f*(*x*) =*ax*^{2}+*bx*+*c*à*f*(*x*) =*a*(*x-h*)^{2}+*k*.

Math is Fun shows the new form in the graph below:

- and
*k*make the graph’s vertex (*h*,*k*) and show us the lowest point (on a positive parabola as shown) or the highest point (on a negative parabola) on the graph. **Find the**or the point on the graph where the line crosses the*y*-intercept,*y*-axis. Every parabola has one y-intercept that can be identified by the number that takes the place of*c*. The coordinates of the*y*-intercept are (0,*c*).**Find the**or the point(s) on the graph where the parabola crosses the*x*-intercept(s),*x*-axis. A parabola can have zero, one, or two*x*-intercepts. A parabola can cross the*x*-axis zero times if it is in the “rainbow” shape and its vertex is located below the*x*-axis, or if it is in the “smile” shape and its vertex is located above the*x*-axis. A parabola can cross the*x*-axis twice when both sides of the parabola cross the*x*-axis. Or it can have one*x*-intercept when the vertex has a*y*-coordinate of 0. In this case, you already know that the*x*-intercept is the same as the*x*-coordinate of the vertex.

When given a quadratic equation, you can find the *x*-intercepts by using the quadratic formula. When you memorize the quadratic formula, it allows you to solve for when *y*=0, or in other words the two times the line crosses the *x*-axis. You only need to use the quadratic formula when you’ve determined that the line has two *x*-intercepts.

Below is the quadratic formula. By replacing *a*, *b*, and *c* with the corresponding values from the quadratic equation you’re working with, you can solve for *x* and find the *x*-intercepts.

**Practice makes perfect.**Use the above tips to work through a practice problem from MathisFun.com:

- Graph
*f*(*x*) = 2*x*^{2}- 12*x*+ 16. - Note: From the original quadratic equation,
*a*=2,*b*=-12, and*c*=16. - Note: Since
*a*is a positive number, we know that the parabola will be positive. - Calculate
*h*:*h*= -*b*/2*a*= -(-12)/(2 ⋅ 2) =**3** - Calculate
*k*(with the above information):*f*(3) = 2(3)^{2}- 12·3 + 16 = 18-36+16 =**-2** - The vertex is (3,-2) with an axis of
*x*=3. - Find the
*x*- and*y*-intercepts.- The coordinates of the y-intercept are (0,
*c*). From our example, the*y*-intercept is (0, 16). - Using the quadratic formula, we can solve for the
*x*-intercepts:

- The coordinates of the y-intercept are (0,

At this point, you need to solve the equation for *x* in two different ways (indicated by the ±symbol). First, solve it by adding 4 to 12 and calculating the answer.

So 4 is the first *x*-intercept. To get the second *x*-intercept, solve the equation by subtracting 4 from 12 and caluclate

Answer: *x* = 4 OR *x* = 2

The parabola crosses the *x*-axis at both (4,0) and (2,0).

- Our graph will take the following shape:

**Continue your practice for the TASC test by taking our quadratic functions quiz.**