Quadratic Equations and Functions
Graph Quadratic Functions Using Properties
Learning Objectives
By the end of this section, you will be able to:
- Recognize the graph of a quadratic function
- Find the axis of symmetry and vertex of a parabola
- Find the intercepts of a parabola
- Graph quadratic functions using properties
- Solve maximum and minimum applications
Before you get started, take this readiness quiz.
Recognize the Graph of a Quadratic Function
Previously we very briefly looked at the function , which we called the square function. It was one of the first non-linear functions we looked at. Now we will graph functions of the form if We call this kind of function a quadratic function.
A quadratic function, where a, b, and c are real numbers and is a function of the form
We graphed the quadratic function by plotting points.
Every quadratic function has a graph that looks like this. We call this figure a parabola.
Let’s practice graphing a parabola by plotting a few points.
Graph
We will graph the function by plotting points.
| Choose integer values for x,
substitute them into the equation and simplify to find . Record the values of the ordered pairs in the chart. |
|
| Plot the points, and then connect
them with a smooth curve. The result will be the graph of the function . |
Graph .
Graph
All graphs of quadratic functions of the form f (x) = ax2 + bx + c are parabolas that open upward or downward. See (Figure).
Notice that the only difference in the two functions is the negative sign before the quadratic term (x2 in the equation of the graph in (Figure)). When the quadratic term, is positive, the parabola opens upward, and when the quadratic term is negative, the parabola opens downward.
For the graph of the quadratic function f (x) = ax2 + bx + c, if
Determine whether each parabola opens upward or downward:
ⓐⓑ
ⓐ
| Find the value of “a”. | |
| Since the “a” is negative, the parabola will open downward. |
ⓑ
| Find the value of “a”. | |
| Since the “a” is positive, the parabola will open upward. |
Determine whether the graph of each function is a parabola that opens upward or downward:
ⓐⓑ
ⓐ up; ⓑ down
Determine whether the graph of each function is a parabola that opens upward or downward:
ⓐⓑ
ⓐ down; ⓑ up
Find the Axis of Symmetry and Vertex of a Parabola
Look again at (Figure). Do you see that we could fold each parabola in half and then one side would lie on top of the other? The ‘fold line’ is a line of symmetry. We call it the axis of symmetry of the parabola.
We show the same two graphs again with the axis of symmetry. See (Figure).
The equation of the axis of symmetry can be derived by using the Quadratic Formula. We will omit the derivation here and proceed directly to using the result. The equation of the axis of symmetry of the graph of f (x) = ax2 + bx + c is
So to find the equation of symmetry of each of the parabolas we graphed above, we will substitute into the formula
Notice that these are the equations of the dashed blue li