Fireworks – Vertex Form of a Quadratic Equation

• Recall that the standard form of a quadratic equation is

y = a·x2 + b·x + c

where a, b, and c are numbers and a does not equal 0.

• The vertex form of a quadratic equation is

y = a·(x – h)2 + k

where (h, k) are the coordinates of the vertex of the parabola and a is a number that does not equal 0.

Fireworks – Vertex Form of a Quadratic Equation

• Vertex form y = a·(x – h)2 + k allows us to find vertex of the parabola without graphing or creating a x-y table.

y = (x – 2)2 + 5 a = 1vertex at (2, 5)

y = 4(x – 6)2 – 3 a = 4vertex at (6, –3)y = 4(x – 6)2 + –3

y = –0.5(x + 1)2 + 9 a = –0.5vertex at (–1, 9)y = –0.5(x – –1)2 + 9

Fireworks – Vertex Form of a Quadratic Equation

• Check your understanding… 1. What are the vertex coordinates of the parabolas with

the following equations?a. y = (x – 4)2 + 1

b. y = 2(x + 7)2 + 3

c. y = –3(x – 5)2 – 12

vertex at (4, 1)

vertex at (–7, 3)

vertex at (5, –12)

2. Create a quadratic equation in vertex form for a "wide" parabola with vertex at (–1, 8).

y = 0.2(x + 1)2 + 8

Fireworks – Vertex Form of a Quadratic Equation

• Finding the a value.

• Recall that the vertex form of a quadratic equation is

y = a·(x – h)2 + k

where (h, k) are the coordinates of the vertex of the parabola and a is a number that does not equal 0.

Also, the values of x and y represent the coordinates of any point (x, y) that is on the parabola.

• We can see that (2, 9) is a point on y = (x – 4)2 + 5

9 = (2 – 4)2 + 59 = 4 + 59 = 9 …because the equation is true

Fireworks – Vertex Form of a Quadratic Equation

• Finding the a value (cont'd)• If we know the coordinates of the vertex and some other

point on the parabola, then we can find the a value.

• For example,

What is the a value in the equation for a parabola that has a vertex at (3, 4) and an x-intercept at (7, 0)?

y = a·(x – h)2 + k0 = a·(7 – 3)2 + 40 = a·(4)2 + 40 = a·16 + 4-4 = a·16

-0.25 = a

substitutesimplify

simplifysubtract 4divide by 16y = -0.25·(x – 3)2 + 4

Fireworks – Vertex Form of a Quadratic Equation

• Finding the a value (cont'd)

What is the a value in the equation for a parabola that has a vertex at (2, -10) and other point at (3, -15)?

Fireworks – Vertex Form of a Quadratic Equation

• Classwork assignment

A particular parabola has its vertex at (-3, 8) and an x-intercept at (1, 0). Your task is to determine which of the following are other points on that same parabola.

1. (-1, 6)

2. (0, 3)

3. (4, -16)

4. (5, -24)