Quadratic Functions 3'1

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Quadratic Functions3.1
Mac 1140 Joan Kessler
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Graphs of Quadratic Functions
The graph of any quadratic function is called a
parabola. If the coefficient of x2 is
positive, the parabola opens upward If the
coefficient of x2 is negative the parabola opens
downward. The vertex (or turning point) is the
minimum or maximum point.
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In calculus you will learn why this is
true. Which of the following open up? y 3x2
- 9 y 73x - x2 y (3 - x)2 8
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The Standard Form of a Quadratic Function

• The quadratic function
• f (x) a(x - h)2 k, a ? 0
• is in standard form.
• NOTE f(x) ax2 bx c is called the
  general form
• The graph of f(x) is a parabola whose vertex is
  the point (h, k).

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f(x) a(x - h)2 k, a ? 0

• The parabola
• - is symmetric to the line x h.
• - opens upward if a gt 0
• opens downward if a lt 0
• has vertex at (h, k)

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f(x) 3(x - 1)2 5,
Ex. 1

• This parabola
• - is symmetric to the line x 1.
• - opens upward
• - has vertex at (1, 5)

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(No Transcript)
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• Graph the quadratic function
• f(x) -2(x - 3)2 8
• without a calculator

Step 1 Determine how the parabola opens.
Note that a , the coefficient of x 2, -2.
Since a lt 0 the parabola opens downward
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Step 2 Find the vertex of f (x) -2(x - 3)2
8. The vertex of the parabola is at (h, k).
Because h 3 and k 8, the parabola has its
vertex at (3, 8).
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Step 3 Find the x-intercepts. Replace f
(x) with 0 in f (x) -2(x - 3)2 8.
0 -2(x - 3)2 8 Find
x-intercepts, setting f (x) equal to zero.
(x - 3)2 4
(x - 3) 2
x - 3 -2 or x - 3 2
x 1 or x 5
The x-intercepts are 1 and 5. The parabola
passes through (1, 0) and (5, 0).
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Step 4 Find the y - intercept. Replace x
with 0 in f (x) -2(x - 3)2 8.
f (0) -2(0 - 3)2 8 -2(-3)2
8 -10
The y - intercept is -10. The parabola passes
through (0, -10).
Step 5 Graph the parabola knowing The
parabola opens downward
Has its vertex at (3, 8).
The x-intercepts are 1 and 5.
The y - intercept is -10.
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With a vertex at (3, 8), x-intercepts at 1 and
5, and a y-intercept at 10, the axis of
symmetry is the vertical line whose equation is x
3.
vertex
x- intercepts
y- intercept
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Finding the Vertex of a Parabola whose equation
is f (x) ax 2 bx c
Find the vertex of f (x) 2x 2 4x - 5

• The vertex is at

V (-1, ? ). Sub -1 into the function
Vertex (-1, -7)
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Minimum and Maximum Quadratic Functions

• Consider f(x) ax2 bx c.
• If a gt 0, then f has a minimum at the vertex
  
• If a lt 0, the f has a maximum at the vertex.

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Example A fence is to be built to form a
rectangular corral along the side of a barn. If
120 feet of fencing are available, what are the
dimensions of the corral of maximum area?
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Let x represent the width of the corral and 120
2x the length.
Area LW (120 2x) x 2x2 120 x
Since the graph of the function is a parabola
and opens downward, we know this function has a
maximum.
Find the length now. 120 2x 120 2(30) 60
The maximum area occurs when the width is 30 feet
and the length is 60 feet.
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Example A basketball is thrown from the free
throw line from a height of six feet. What is the
maximum height of the ball if the path of the
ball is
The path is a parabola opening downward. The
maximum height occurs at the vertex.
So, the vertex is (9, 15).
The maximum height of the ball is 15 feet.
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Using a Quadratic Model
Use the calculator, to find a quadratic
regression equation. Then use the equation to
find the maximum price.
Enter the data into L1, L2Stat, Edit
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Using a Quadratic Model
Use the calculator, to find a quadratic
regression equation. Then use the equation to
find the maximum price for the model. STAT CALC 5
Press y Make sure Plot1 is highlighted
Press 2nd STATPLOT Enter

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Using a Quadratic Model
Use the calculator, to find a quadratic
regression equation. Then use the equation to
find the maximum price for the model.
Press Zoom 9

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Another Example quadratic model
Use the calculator, to find a quadratic
regression equation. Then use the equation to
find the age for which people spend the most
money on groceries. What is the expected cost per
month at age 65?
y -. 28x218.94x139.62

age 65 181.40
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You can find handouts for learning how to use
your calculator on the class websitewww.distance
math.com/collegeacademy