For this section of the lesson, you will explore characteristics of the graph of a quadratic function if the equation is written in vertex form.

Vertex Form: y = a(x − h)² + k

Investigating the Vertex Form of a Quadratic Equation

1. Click on the link below to access the applet. Investigate the vertex form of a quadratic equation.
2. Use the scroll bar in the applet to slide the values of a, h, and k. Observe how these values change the parabola.

   Role of a, h and k in y = a (x - h)²+ k

3. How did a affect the graph? Once you have formulated a conjecture, complete the following statements.
   • If a < 0 (or negative), then the parabola . . . Interactive popup. Assistance may be required. Check Your Answer faces down.
   • If a > 0 (or positive), then the parabola . . . Interactive popup. Assistance may be required. Check Your Answer faces up.
4. Record what you observe.
5. What other observations did you make?
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• 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.

6. How did h and k affect the graph?
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• h shifts the parabola right or left. If the expression reads (x – 5), this is a shift of 5 right.
• k shifts the parabola up and down.
• The vertex of the parabola is value of (h, k).

Name the Vertex for the Following Quadratic Functions

1. y = (x − 2)² + 6
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(2, 6)
2. y = - 2(x + 5)² + 4
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(-5, 4)
3. y = - (x − 3)² + 1
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(3, 1)

Complete the following statement. (Click or touch the blanks to see the correct answers.)

When the equation of a quadratic function is written in vertex form: y = a(x − h)² + k, the Interactive button. Assistance may be required. __________ vertex can easily be identified by Interactive button. Assistance may be required. __________ (h,k) .

Axis of Symmetry

In previous lesson, we learned that the axis of symmetry runs down the center of the parabola through the vertex.

Therefore the axis of symmetry for equation 1 is x = 2. (Remember that 2 is the x-coordinate of the vertex.)

Steps to Find the y-intercept of a Quadratic Function Written in Vertex Form

Do you remember how you found the y-intercept of a linear function?

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Quick Recap

Look at the y-intercept of the graph on the right.

Recall that the x-value of the y-intercept is always 0. Therefore, to get the y-intercept from the equation, substitute 0 for x and solve for y.

Example:

2x − 3y

= 6

Step 1. Substitute 0 for x:

2(0) − 3y

= 6

Step 2: Solve for y:

-3y
y
or

= 6
= -2
(0, -2)

Since the y-intercept of a quadratic function also has an x-value of 0, you will use the same process.

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

I'll do this one.

You do this one.

y = 2(x − 3)² –10

y = -3(x + 1)² + 7

Since a > 0, I know the parabola is facing up.

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Parabola is facing down.

The vertex is (3, -10) since (h, k) represents the vertex.

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Vertex is (-1, 7)

The axis of symmetry is x = 3 since the x-coordinate of the vertex is 3.

y = 2(0 − 3)² − 10
y = 2(-3)² – 10
y = 8
The y-intercept is (0, 8).

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The axis of symmetry is x = -1

To find the y-intercept, I will substitute 0 in for x and solve for y.

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The y-intercept is (0, 4).

Since a = 2, I also know that the graph is facing up and has a vertical stretch of 2.

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Check Your Answer

Parabola is facing down and has a vertical stretch of 3.