1. Two forms of a quadratic equation2. Review: Graphing using transformations3. Properties of the graph4. Graphing by hand

a) Method 1: Use standard formComplete the Square

b) Method 2: Use quadratic form:5. Identify the vertex and axis of symmetry6. Find maximum or minimum of applied quadratic problems

Section 3-1 Quadratic Functions

f (x) a(x h)2 k

cbxaxxf 2)(

1. Two forms of a quadratic Functions

1. A quadratic function is a function of the form:

where a, b, and c are real numbers and a ≠ 0.

cbxaxxf 2)(

2. Standard (Vertex) Form of a quadratic function:

f (x) a(x h)2 k

1)3(2)( 2 xxf

2. Review: Graphing using Transformations

1. Identify the x and y intercepts

2. Identify the parent function

3. Describe the sequence of transformations

4. graph

5. Does this function have a maximum or minimum? What is it?

0a

0a

Opens up

Opens down

3. Properties of the graph

Axis of Symmetry

Vertex

Vertex

Axis of Symmetry

Maximum at vertex

Minimum at vertex

Parabola

Method 1: Graph standard (vertex) form of quadratic:

Direction of parabola: If a> 0 then graph opens up; If a<0 then graph opens down

Vertex :

x-intercepts: Solve:

y-intercept:

Axis of Symmetry: (points on the graph are equidistant horizontally from x=h)

f (x) a(x h)2 k

f (x) a(x h)2 k

),( kh

0)( xf))0(,0( f

hx

4)1(2)( 2 xxfPractice:

4 a)Complete the Square

Complete the square of

to rewrite in standard form:

f (x) ax 2 bx c

f (x) a(x h)2 k

Determine the vertex and axis of symmetry for the quadratic:

16)( 2 xxxf

4 b) Method 2: Find vertex directly

Axis of Symmetry:

Vertex:

Find y-interceptsFind the x-intercepts.

Additional point using symmetry.

x b2a

a

bf

a

b

2,

2

f (x) ax 2 bx c

f (x) ax 2 bx c

a > 0: opens upa < 0: opens down

Let’s try the previous example again, using Method 2.

5. The DiscriminantWhat does the discriminant tell you? # x-intercepts 042 acb

042 acb

042 acb

16)( 2 xxxf

Example: Two Ways to Graph Quadratic Functions by hand

Which method do you prefer for graphing:

f (x) 3x 2 12x 1

6. a) Determine the quadratic equation:Given the vertex and a point

Determine the quadratic equation whose vertex is (4,-1) and passes through the point (2,7).

Recall standard form:

f (x) a(x h)2 k

6 b) Determine the quadratic equation:Given x-intercepts and a point

Suppose and r1 and r2 are x-intercepts.

We can write

Example: Suppose a quadratic has zeros at -3 and 5. And suppose the function passes through the point (6,63). Write the quadratic.

042 acb

))(()( 21 rxrxaxf

7. Properties of the graph.

Max/Min is at: and

Domain:

Range: If a > 0, then Range= If a < 0, then Range=

Increasing/Decreasing:

f (x) 2x 2 8x 3

Minyy |

Maxyy |

a

bfMaxMin

2/

a

bx

2

ApplicationExample: A farmer has 2000 ft of fencing to enclose a rectangular area that borders a river. The fencing will be along the 3 sides other than the river.

a) Express area as a function of one variable.

b) Determine the dimensions that will maximize the area.

Maximums and MinimumsAn engineer collects the following data showing the speed s of a Ford Taurus and its average miles per gallon M.

Speed, s Miles/Gallon, M

30 18

35 20

40 23

40 25

45 25

50 28

55 30

60 29

65 26

65 25

70 25

a) Determine a scatter plot for the data

b) Use a graphing utility to find the quadratic function of best fit to this data

c) Use the function found to determine the speed that maximizes miles per gallon

d) Use the function to predict miles per gallon for a speed of 63 mph.

Section 3.1p. 164# 29,35,42,49,53,55,59,61,67,(69),71,78,79,81