This lesson is about using an equation of a quadratic function to find the characteristics of its graph. You will need a graphing calculator for this lesson.

Recall that the standard form of a quadratic equation is y = ax2 + bx + c where a is the coefficient of x2 and b is the coefficient of x and c is a constant.

Quadratic Functions (General Form) Exercise

  1. This activity might not be viewable on your mobile device. Interactive exercise. Assistance may be required. Use the applet below to find a, b, and c and to explore how a, b, and c change the graph of a quadratic function.
  2. This activity might not be viewable on your mobile device. Interactive exercise. Assistance may be required. Quadratic Functions Click on the link to access the applet. Then click on the draw button to begin the interactive. Next, drag the button above a, b, and c. Observe how the parabola changes as the value of a, b, and c changes.
  3. In your notes, write down what you observe.
  4. What were your observations?
    5. Interactive popup. Assistance may be required.

    Check Your Answer

    Possible Observations
    • If a < 0 (or negative), then the parabola faces down.
    • If a > 0 (or positive), then the parabola faces up.
    • As the value of a gets closer to 0, the parabola gets wider. Or as the │a│ decreases, the parabola gets wider. This is a vertical compression.
    • As the value of a gets farther from 0, the parabola gets thinner. Or as the │a│ increases, the parabola gets wider. This is a vertical stretch.
    • c is the y-intercept.
    Close Pop Up

Graphs of Quadratic Functions Exercise

  1. This activity might not be viewable on your mobile device. Interactive exercise. Assistance may be required. Follow this link to an applet where you can observe how changing the b value of the equation changes the graph: Investigating Graphs of Quadratic Functions
  2. Slide the dot on the b line.
  3. Use the Take Notes tool to write what you observe.

Flowchart to Find Vertex and Axis of Symmetry of a Quadratic Function
Given the Standard Form

Steps

Example

Standard form: y = ax2 + bx + c

Example: y = 2x2 − 12x + 14

First find x by using the formula x = -b 2a .

(This tells us to tak\e the opposite of b and divide it by the quantity of 2 times a.)

First find x by using x = 12 2(2) = 3.

Next substitute the x-value into the original equation to find y.

y = 2(3)2 − 12(3) + 14 = -4
or y = -4

Vertex is (3, -4)

Since the axis of symmetry for a quadratic function is a vertical line that runs through the vertex, then the equation for the axis of symmetry is x = x-coordinate of vertex.

Axis of symmetry: x = 3

Follow the link below to see how to graph a quadratic function whose equation is in standard form.

Equation of a Parabola

Now, let’s put all the characteristics together to determine the parts of the parabola.

I'll do this one...

You do this one...

y = - 2x2 + 4x − 5

y = x2 + 4x + 7

Since a < 0, I know the parabola is facing down.
Interactive popup. Assistance may be required.

Check Your Answer

The paraboloa is facing up.Close Pop Up

To find the x-coordinate of vertex, I will use

x = -b 2a

x = -4 2(-2) = 1
Interactive popup. Assistance may be required.

Check Your Answer

x-coordinate of vertex: x = -4 2(1) = -2 Close Pop Up

To find the y-coordinate of vertex, I will substitute 1 in for x in the original equation.

y = - 2(1)2 + 4(1) − 5 = - 3

Vertex: (1, - 3)
Interactive popup. Assistance may be required.

Check Your Answer

y-coordinate: y = (- 2)2 + 4(- 2) + 7 = 3
Vertex: (- 2, 3)Close Pop Up

Since the x-coordinate of the vertex is 1, I know that the equation for the axis of symmetry is x = 1.
Interactive popup. Assistance may be required.

Check Your Answer

Axis of symmetry: x = - 2Close Pop Up

Since c = -5, I know this is the y-intercept.
Interactive popup. Assistance may be required.

Check Your Answer

The y-intercept is 7.Close Pop Up