Quadratics

Parts of a Parabola and Vertex Form

Quadratic Functions

Function:

Standard Form (Vertex Form):

Graphs a parabola

All are symmetric to a line called the axis of symmetry or line of symmetry (los)

2( )f x ax bx c

2( ) ( )f x a x h k

Show that g represents a quadratic function. Identify a, b, and c.

723)( xxxg

14193

142213

723)(

2

2

xx

xxx

xxxg

14

19

3

c

b

a

Show that g represents a quadratic function. Identify a, b, and c.

2582)( xxxg

327610

3280410

25162

2582)(

2

2

xx

xxx

xx

xxxg

32

76

10

c

b

a

Show that g represents a quadratic function. Identify a, b, and c.

46)( 2 xxg

3212

43666

466

46)(

2

2

2

xx

xxx

xx

xxg

32

12

1

c

b

a

Parts of a Parabola

Axis of Symmetry (Line of Symmetry) LOS:

The line that divides the parabola into two parts that are mirror images of each other.

Vertex: Either the lowest or highest point.

Let’s look at a transformation

2)( xxf

UpConcave

Parent

23)( 2 xxf

UpConcave

FormVertex

3.Rt

2Up

What is the vertex, max or min, and los?

23)( 2 xxfVertex Form:

)2,3(:Vertex

2:min y

3: xlos

Let’s look at a transformation

2)( xxf

UpConcave

Parent

24)( xxf

UpConcave

FormVertex

4LeftWhat is the vertex, max or min, and los?

24)( xxfVertex Form:

)0,4(: Vertex

0:min y

4: xlos

Let’s look at a transformation

2)( xxf

UpConcave

Parent

2)0()( 2 xxf

DownConcave

FormVertex

Flips

2Down

What is the vertex, max or min, and los?

2)( 2 xxf

Vertex Form:

)2,0(: Vertex

2:min y

0: xlos

Finding the Vertex and los on the GDC

Put equation into y1. Press Zoom 6 – fix if necessary by changing

the window. Press 2nd Trace (Calc). Press max/min. Left of it; Then right of it; Then ENTER.

Let’s Try It….

Find the vertex, min, and los.

473)( 2 xxxf

17.1:

08.8:min

08.8,17.1:

xlos

y

Vertex

Use your GDC to find the zeros (x-intercepts)….

Press 2nd Trace (Calc) Press zero. Again, to the left, to the right, ENTER.

Find the zeros for the last example.

473)( 2 xxxf

47.0

8.2

x

x

Example:

Find the vertex, los, max/min, zeros, and tell whether concave up or concave down.

832)( 2 xxxf

DownConcave

xxzeros

y

xlos

Vertex

89.2,39.1:

13.9:max

75.0:

13.9,75.0:

Example:

Find the vertex, los, max/min, zeros, and tell whether concave up or concave down.

523)( 2 xxxf

UpConcave

xxzeros

y

xlos

Vertex

1,70.1:

3.5:min

3.0:

3.5,3.0:

These are the solutions.What are the x-intercepts?

Example:

Find the vertex, los, max/min, zeros, and tell whether concave up or concave down.

762)( 2 xxxf

DownConcave

SolutionsREALNoSolutions

NONEx

y

xlos

Vertex

:

:int

5.2:max

5.1:

5.2,5.1:

These are the solutions.What are the x-intercepts?

Example

Find the vertex, los, max/min, zeros, and tell whether concave up or concave down.

235)( 2 xxxf

UpConcave

Nonezeros

y

xlos

Vertex

:

55.1:min

30.0:

55.1,30.0:

Example:

Find the vertex, los, max/min, zeros, and tell whether concave up or concave down.

xxxf 7)( 2

DownConcave

xxzeros

y

xlos

Vertex

7,0:

25.12:max

5.3:

25.12,5.3:

Example:

Find the vertex, los, max/min, zeros, and tell whether concave up or concave down.

2023)( 2 xxxf

UpConcave

xxzeros

y

xlos

Vertex

3.2,9.2:

3.20:min

3.0:

3.20,3.0:

How do I know if it is concave up or down just by looking at the function?

In the following examples, state whether the parabola is concave up or down and whether the vertex is a max or a min by just looking at the function.

max,DownConcave

178)( 2 xxxf

147)( 2 xxxf

xxxf 238)(

262)( xxxf

min,UpConcave

max,DownConcave

min,UpConcave

Write the equation in standard form of the parabola whose vertex is (1, 2) and passes through the point (3, -6).

khxay 2)(

(h, k)

(x, y)

2

48

2136 2

a

a

a

212

:2 xy

FORMSTANDARD

Write the equation in standard form of the parabola whose vertex is (-2, -1) and passes through the point (0, 3).

khxay 2)(

(h, k)(x, y)

1

44

1203 2

a

a

a

12

:2 xy

FORMSTANDARD

Write the equation in standard form of the parabola whose vertex is (4, -1) and passes through the point (2, 3).

khxay 2)(

(h, k)

(x, y)

1

44

1423 2

a

a

a 14

:2 xy

FORMSTANDARD

A golf ball is hit from the ground. Its height in feet above the ground is modeled by the function where t represents the time in seconds after the ball is hit.

How long is the ball in the air?

What is the maximum height of the ball?

,18016 2 ttth

Graph on GDC.Find the zeros. Answer: 11.25 seconds

Graph on GDC.Find the maximum y-value.

Answer: 506.25 feet

A. What is the maximum height of the ball?

B. At what time does the ball reach its maximum height?

C. At what time(s) is the ball 16 feet high in the air?

Graph and find the maximum y-value.

Answer: 21 feet

Set equation = to 21and find the zeros. Answer: 1 second

Set equation = to 16and find the zeros. Answer: 1.56 seconds and

0.44 seconds.

. A. What ticket price gives the maximum profit?

B. What is the maximum profit?

C. What ticket price would generate a profit of $5424?

Graph and find the maximum x-value.Hint: Press Zoom 0 and change x-max to 50and y max to 8000.

Answer: $25

Answer: $6,000

Set equation = to 5424and find the zeros.Hint: Press zoom 0.

Answer: $19 or $31

Graph and find the maximum y-value.Hint: Press Zoom 0 and change x-max to 50and y max to 8000.