Standard 9 Write a quadratic function in vertex form

Vertex form- Is a way of writing a quadratic equation that facilitates finding the vertex.

y – k = a(x – h)2

The h and the k represent the coordinates of the vertex in the form V(h, k).The “a” if it is positive it will mean that our parabola opens upward and if negative it will open downward.A small value for a will mean that our parabola is wider and vice versa.

Standard 9 Write a quadratic function in vertex form

Write y = x2 – 10x + 22 in vertex form. Then identify the vertex.

y = x2 – 10x + 22 Write original function.

y + ? = (x2 –10x + ? ) + 22 Prepare to complete the square.

y + 25 = (x2 – 10x + 25) + 22Add –102

2( ) = (–5)2= 25 to each side.

y + 25 = (x – 5)2 + 22 Write x2 – 10x + 25 as a binomial squared.

y + 3 = (x – 5)2 Write in vertex form.

The vertex form of the function is y + 3 = (x – 5)2. The vertex is (5, –3).

ANSWER

EXAMPLE 7 Find the maximum value of a quadratic function

The height y (in feet) of a baseball t seconds after it is hit is given by this function:

Baseball

y = –16t2 + 96t + 3

Find the maximum height of the baseball.

SOLUTION

The maximum height of the baseball is the y-coordinate of the vertex of the parabola with the given equation.

EXAMPLE 7 Find the maximum value of a quadratic function

y = –16t2 + 96t +3 Write original function.

y +(–16)(?) = –16(t2 –6t + ? ) + 3 Prepare to complete the square.

y – 144 = –16(t – 3)2 + 3 Write t2 – 6t + 9 as a binomial squared.

y – 147 = –16(t – 3)2 Vertex Form

y = –16(t2 – 6t) +3 Factor –16 from first two terms.

y +(–16)(9) = –16(t2 – 6t + 9 ) + 3 Add to each side.(–16)(9)

The vertex is (3, 147), so the maximum height of the baseball is 147 feet.

ANSWER

GUIDED PRACTICE for Examples 6 and 7

y = x2 – 8x + 17

y - 1 = (x – 4)2 ; (4, 1).ANSWER

13.

Write the quadratic function in vertex form. Then identify the vertex.

y = x2 + 6x + 3

y + 6 = (x + 3)2 ; (–3, –6)

ANSWER

14.

f(x) = x2 – 4x – 4

y + 8 = (x – 2)2 ; (2 , –8)ANSWER

15.

GUIDED PRACTICE for Examples 6 and 7

16. What if ? In example 7, suppose the height of the baseball is given by y = – 16t2 + 80t + 2. Find the maximum height of the baseball.

102 feet.ANSWER

EXAMPLE 1 Write a quadratic function in vertex form

Write a quadratic function for the parabola shown.

SOLUTION

Use vertex form because the vertex is given.

y – k = a(x – h)2 Vertex form

y = a(x – 1)2 – 2 Substitute 1 for h and –2 for k.

Use the other given point, (3, 2), to find a.2 = a(3 – 1)2 – 2 Substitute 3 for x and 2 for y.

2 = 4a – 2 Simplify coefficient of a.

1 = a Solve for a.

EXAMPLE 1 Write a quadratic function in vertex form

A quadratic function for the parabola is y = (x – 1)2 – 2.

ANSWER

EXAMPLE 1 Graph a quadratic function in vertex form

Graph y – 5 = – (x + 2)2.14

SOLUTION

STEP 1 Identify the constants a = – , h = – 2, and k = 5.

Because a < 0, the parabola opens down.

14

STEP 2 Plot the vertex (h, k) = (– 2, 5) and draw the axis of symmetry x = – 2.

EXAMPLE 1 Graph a quadratic function in vertex form

STEP 3 Evaluate the function for two values of x.

x = 0: y = (0 + 2)2 + 5 = 414

–

x = 2: y = (2 + 2)2 + 5 = 114

–

Plot the points (0, 4) and (2, 1) and their reflections in the axis of symmetry.

STEP 4 Draw a parabola through the plotted points.

GUIDED PRACTICE for Examples 1 and 2

Graph the function. Label the vertex and axis of symmetry.

1. y = (x + 2)2 – 3 2. y = –(x + 1)2 + 5

GUIDED PRACTICE for Examples 1 and 2

3. f(x) = (x – 3)2 – 412