Download Section 3-1 Quadratic Functions

1.
2.
3.
4.
Two forms of a quadratic equation
Review: Graphing using transformations
Properties of the graph
Graphing by hand
2
f
(x)

a(x

h)
k
a) Method 1: Use standard form
Complete the Square
2
f
(
x
)

ax
 bx  c
a) Method 2: Use quadratic form:
5. Identify the vertex and axis of symmetry
6. Find maximum or minimum
 of applied quadratic problems
Section 3-1 Quadratic Functions
1. Two forms of a quadratic Functions
1. A quadratic function is a function of the form:
f ( x)  ax 2  bx  c
where a, b, and c are real numbers and a ≠ 0.
2. Standard (Vertex) Form of a quadratic
function:
2
f (x)  a(x  h)  k

2. Review: Graphing using Transformations
f ( x)  2( x  3) 2  1
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?
3. Properties of the graph
a0
Axis of
Symmetry
Opens up
Minimum at vertex
Vertex
Vertex
a0
Opens down
Maximum at vertex
Axis of
Symmetry
Method 1: Graph standard (vertex) form of quadratic:
f (x)  a(x  h) 2  k
f (x)  a(x  h)  k
2
Direction
of parabola: If a> 0 then graph opens up;

If a<0 then graph opens down
 : ( h, k )
Vertex
f ( x)  0
y-intercept: (0, f (0))
x-intercepts: Solve:
Axis of Symmetry: x  h
(points on the graph are equidistant
horizontally from x=h)
Practice:
f ( x)  2( x  1)  4
2
4 a)Complete the Square
Complete the square of
to rewrite in standard form:
f (x)  ax  bx  c
2
f (x)  a(x  h)  k
2

Determine the vertex and axis of symmetry for the quadratic:
f ( x)  x  6
x  1
2
4 b) Method 2: Find
vertex directly
2
f (x)  ax  bx  c
f (x)  ax 2  bx  c
a > 0: opens up
 down
a < 0: opens
Axis of Symmetry:
x
b
2a
Vertex:   b   b  
 
, f
 
 2a  2a  
Find y-intercepts
Find the x-intercepts.
Additional point using symmetry.
5. The Discriminant
What does the discriminant tell you?
# x-intercepts
b 2  4ac  0
b 2  4ac  0
b 2  4ac  0
Let’s try the previous example again, using Method 2.
f ( x)  x 2  6 x  1
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)  k
2
6 b) Determine the quadratic equation:
Given x-intercepts and a point
2
b
 4ac  0
Suppose
We can write
and r1 and r2 are x-intercepts.
f ( x)  a( x  r1 )( x  r2 )
Example:
Suppose a quadratic has zeros at -3 and 5. And suppose the
function passes through the point (6,63).
Write the quadratic.
7. Properties of the graph.
f (x)  2x 2  8x  3
b
Max/Min is at: x 
and Min / Max 
2a
Domain:
Range: If a > 0, then Range=
If a < 0, then Range=
Increasing/Decreasing:
y | y  Min
y | y  Max
 b
f

 2a 
Application
Example: 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 Minimums
An engineer collects the following data showing the speed s of
a Ford Taurus and its average miles per gallon M.
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.
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
Section 3.1
p. 164
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