Graphing Quadratic Functions | Mathematics
A quadratic function is in the form of f(x) = ax2 + bx + c. The squared exponent (x2) 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/2a and k = f(h).
- Rearrange f(x) = ax2 + 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 y-intercept, or the point on the graph where the line crosses the 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 x-intercept(s), or the point(s) on the graph where the parabola crosses the 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) = 2x2 - 12x + 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/2a = -(-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:
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.