Transcript Quadratic Functions

Quadratic Functions;
An Introduction
Mr. J. Grossman
Graph the function y = x²
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Describe the graph.
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What does it look like?
Graph the function y = x²
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The graph is…
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Nonlinear
U-shaped smooth curve
(called a parabola)
Symmetry
Axis of Symmetry: the
line that divides the
parabola into two
matching halves.
It has a vertex.
Graph the function y = x²
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The function we have graphed and described
is known as a Quadratic Function.
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A Quadratic Function is any function that can
be written in the standard form:
y = ax² + bx + c
where a, b, and c are real numbers and a ≠ 0.
You must be able to recognize a Quadratic
Function
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From a graph…
You must be able to recognize a Quadratic
Function
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From a table
of values…
x
y = x^2
0
0
1st Common
Difference
2nd Common
Difference
1
Constant change
in x-values
1
1
2
2
4
3
3
9
5
2
4
16
7
2
You must be able to recognize a Quadratic
Function
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A quadratic
function will
not have a
common first
difference. It
will have a
common
second
difference.
This is true of
all quadratic
functions.
x
y = x^2
0
0
1st Common
Difference
2nd Common
Difference
1
Constant change
in x-values
1
1
2
2
4
3
3
9
5
2
4
16
7
2
You must be able to recognize a Quadratic
Function
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From an equation…
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Must be a 2nd degree polynomial
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y = 2x²
y = -5x² + 6x
f(x) = x² + 9
f(x) = ½ x² + 3x -11
Tell whether each function is a Quadratic
Function:
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{(-4, 8), (-2, 2), (0, 0), (2, 2), (4, 8)}
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y = -3x + 20
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y + 3x² = -4
Quadratic Functions
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Some open upwards; others, downward.
Quadratic Functions
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When a quadratic
function is written in the
form y = ax² + bx + c,
the value of a
determines the
direction in which the
parabola opens.
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If a > 0, the parabola
opens upward.
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If a < 0, the parabola
opens downward.
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Remember, a ≠ 0 !!!
Quadratic Functions
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Graph the function:
f(x) = -4x² - x + 1
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Graph the function:
y – 5x² = 2x – 6
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Does the parabola
open upward or
downward?
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Does the parabola
open upward or
downward?
Quadratic Functions
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The highest or lowest point on the parabola is
known as the vertex.
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If a > 0, the parabola opens upward and the y-value
of the vertex is the minimum value of the function.
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If a < 0, the parabola opens downward and the yvalue of the vertex is the maximum value of the
function.
Quadratic Functions
Quadratic Functions
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The value of a (coefficient of the x² term)
affects the width of the parabola also.
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The smaller the absolute value of a, the wider the
parabola.
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Compare the graphs of: y = -4x², y = ¼ x² + 3, and
y = x².
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Which graph (quadratic function) is widest? Narrowest?
Quadratic Functions
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Recall that the quadratic function in standard form is
written: y = ax² + bx + c.
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The value of c (constant of the quadratic function)
translates the graph of the function up or down the
axis of symmetry.
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Compare the graphs of the following quadratic
functions: y = 2x², y = 2x² + 3, and y = 2x² - 3.
Quadratic Functions
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Positive values of c
shift the vertex UP.
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Negative values of c
shift the vertex DOWN.
Quadratic Functions
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DOMAIN: Unless a specific domain is given,
you may assume that the domain of a
quadratic function is the set of all real
numbers.
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RANGE: The range begins (albeit minimum
or maximum value) with the vertex.
Any questions?
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If not, complete Study Guide Practice 10-1,
pg 128, all problems.