Quadratic Functions
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Anonymous Anonymous
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35 Slides • 17 Questions
1
4.1 Intro to Quadratic Functions
Parabola
2
Examples of Quadratic Functions
3
Multiple Choice
Which of the following is not a quadratic function?
f(x)=x2
f(x)=−2x2+5x+1
f(x)=3x2−9
f(x)=4x−1
f(x)=4x2−x
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Axis of Symmetry
5
Fill in the Blanks
Type answer...
6
Fill in the Blanks
Type answer...
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Fill in the Blanks
Type answer...
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Vertex
Maximum when a < 0
Minimum when a > 0
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How to find the vertex
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Multiple Choice
Find the vertex of
f(x)=x2+2x−3(-1, -4)
(-1, -6)
(1, 0)
(-2, -3)
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Fill in the Blanks
Type answer...
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Summary
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Quadratic Function
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Objectives:
Define and illustarte quadratic equation and quadratic function.
Represent a quadratic function using: (a) table of values; (b) graph; and (c) equation
15
Brain Break
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Quadratic Equation and Quadratic Function
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Example 1: Determine whether the given equation is a quadratic function or not.
18
Example 1: Determine whether the given equation is a quadratic function or not.
19
Multiple Choice
True or False.
4x2 - 3x + 5 = y is a univariate quadratic equation.
True
False
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Multiple Choice
Which of the following is a quadratic function?
y = 3y2 - 8
f(x) = 92
h(x) = 3 + 2x2 + x
x1= 52x+1
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A table of values can be generated from a quadratic function by substituting the x-values and calculating the values for f(x).
How can the solution(s) to a quadratic equation be found from a table of values?
When looking at a table of values for a quadratic function, the x-intercepts represent the x-values where y = 0. This corresponds to the x-values where f(x) is 0 in function notation.
Representation of Quadratic Functions
A. Table of Values
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Representation of Quadratic Functions
A. Table of Values
Example 2: Given the quadratic functions (a) y = x2 - 1 and
(b) f(x) = x2 - 3x + 2, complete the table of values below.
y = x2 - 1
f(x) = x2 - 3x + 2
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Representation of Quadratic Functions
A. Table of Values
Example 2: Given the quadratic functions (a) y = x2 - 1 and
(b) f(x) = x2 - 3x + 2, complete the table of values below.
y = x2 - 1
f(x) = x2 - 3x + 2
zero/x-intercept
zero/x-intercept
zero/x-intercept
zero/x-intercept
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Multiple Choice
Given the table of values of the quadratic function at the left, what is the missing value?
40
25
20
0
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Multiple Choice
Given the table of values of quadratic function at the left, what are the zeros of function?
0 & 1
1 & -3
1 & - 5
-5 & - 3
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Representation of Quadratic Functions
B. Graph
27
Representation of Quadratic Functions
B. Graph
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Representation of Quadratic Functions
B. Graph
Translation
-is the horizontal and vertical movement of the graph of f(x).
Transformation of f(x) = x2 + k (Vertical Shifts)
If k is positive, the graph of y = x2 is shifted upwards by k units.
If k is negative, the graph of y = x2 is shifted downwards by k units.
29
Representation of Quadratic Functions
B. Graph
Transformation of f(x) = (x - h)2 (Horizontal Shifts)
If h is positive, the graph of y = x2 is shifted h units to the right.
If h is negative, the graph of y = x2 is shifted h units to the left.
30
Representation of Quadratic Functions
B. Graph
Transformation of f(x) = a(x - h)2 + k with vertex (h, k)
The axis of symmetry is x = h.
If a > 0, the parabola opens upward and the minimum function value is k.
If a < 0, the parabola opens downward and the maximum function value is k.
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Representation of Quadratic Functions
B. Graph
Example 3: Given the graphs of quadratic functions below, determine the following:
a) vertex
b) axis of symmetry
c) x-intercept/s
d) y-intercept
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Representation of Quadratic Functions
B. Graph
Example 3: Given the graphs of quadratic functions below, determine the following:
33
Representation of Quadratic Functions
B. Graph
Example 4: Given the graphs of quadratic functions below, determine the following:
a) vertex
b) axis of symmetry
c) x-intercept/s
d) y-intercept
e) function
34
Representation of Quadratic Functions
B. Graph
Example 4: Given the graphs of quadratic functions below, determine the following:
35
Representation of Quadratic Functions
B. Graph
Example 5: Given the graphs of quadratic functions below, determine the following:
a) vertex
b) axis of symmetry
c) x-intercept/s
d) y-intercept
e) function
36
Representation of Quadratic Functions
B. Graph
Example 5: Given the graphs of quadratic functions below, determine the following:
37
Representation of Quadratic Functions
B. Graph
Example 6: Compare the graph of the first equation to the graph of the second equation.
a) y = 3x2, y = x2
Both graphs open upwards. The graph of y = 3x2 is narrower than the graph of y = x2.
38
Representation of Quadratic Functions
B. Graph
Example 6: Compare the graph of the first equation to the graph of the second equation.
c) y = (x - 3)2 + 9, y = x2 + 9
d) y = (x - 5)2 - 4, y = x2
Both graphs open upwards. The value of h in
y = (x - 3)2 + 9 is 3. Hence, its graph is translated 3 units to the right of the graph of
y = x2 + 9.
Both graphs open upwards.
k = -4 in y = (x - 5)2 - 4 while
h = 5 in y = (x - 5)2 - 4. Hence, the graph of y = (x - 5)2 - 4 is translated 4 units downward and 5 units to the right of the graph of y = x2.
39
Multiple Choice
What is the vertex of the function?
(1, 3)
(3, 1)
(2, 4)
(4, 2)
40
Multiple Choice
Which of the following describes the function graph?
y = (x - 1)2 + 3
y = -(x + 1)2 + 3
y = -(x - 1)2 + 3
y = (x - 1)2 - 3
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Multiple Choice
What are the x-intercepts of the function?
1 and -1
-1 and 0
-2 and 0
2 and 0
42
Representation of Quadratic Functions
C. Equation
Forms of Quadratic Functions
General Form :
f(x) = a(x - h)2 + k
Vertex Form :
a :
(h, k) :
x = h :
leading coefficient
vertex
axis of symmetry
f(x) = ax2 + bx + c
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Representation of Quadratic Functions
C. Equation
Vertex of a Parabola
axis of symmetry :
x-coordinate of the vertex:
y-coordinate of the vertex:
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Representation of Quadratic Functions
C. Equation
Example 7: Write y = (x + 3)2 - 5 in quadratic form.
Solution:
y = (x + 3)2 - 5
y = (x + 3)(x + 3) - 5
y = x2 + 3x + 3x + 9 - 5
y = x2 + 6x + 4
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Representation of Quadratic Functions
C. Equation
Example 8: Write y = x2 - 4x - 5 in vertex form. Identify the vertex and the axis of symmetry.
Solution:
y = x2 - 4x - 5 use completing the square
y = (x2 - 4x + 4) - 4 - 5
y = (x2 - 4x + 4) - 9
y = (x - 2)2 - 9
vertex: (2, -9)
axis of symmetry: x = 2
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Representation of Quadratic Functions
C. Equation
Example 9: Identify the vertex and the axis of symmetry of
y = -x2 + 2x + 2.
Solution:
a = -1, b = 2, c = 2
axis of symmetry: x = 1
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Representation of Quadratic Functions
C. Equation
Example 9: Identify the vertex and the axis of symmetry of
y = -x2 + 2x + 2.
Alternative solution:
a = -1, b = 2, c = 2
axis of symmetry: x = 1
48
Multiple Choice
Identify the axis of symmetry of
y = x2 - 4x - 8
1
2
-4
8
49
Multiple Choice
Identify the vertex of
y = x2 - 4x - 8
(-2, 4)
(4, 8)
(2, -12)
8, -12)
50
Multiple Choice
Which of the following is the general form of y = (x - 1)2 + 3?
y = x2 - 2x + 4
y = x2 + 2x + 2
y = x2 - 2x - 4
y = x2 - 2x - 2
51
Multiple Choice
Which of the following is the vertex form of y = x2 - 6x + 8?
y = (x 3)2 - 17
y = (x - 3)2 + 1
y = (x + 3)2 - 17
y = (x - 3)2 - 1
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Worktext Reference:
Oronce O.A., Mendoza M.O. (2019). E-Math Worktext in Mathematics 9, Manila, Philippines: REX Bookstore Inc.
Reference:
Nivera G.C., Lapinid M.R.C. (2018). Grade 9 Mathematics Patterns and Practicalities 9, Makati City, Philippines: Salesiana Books by Don Bosco Press, Inc.
Albay E.M., Oli M.C. (2018). Practical Math 9, Makati City, Philippines: Diwa Learning Systems Inc.
Web Reference:
https://www.texasgateway.org/resource/solving-quadratic-equations-using-tables#section-id-138421
Digital Graphing Tool:
4.1 Intro to Quadratic Functions
Parabola
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