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```3.1 – Quadratic Functions and Application
A function defined by f(x) = ax2 + bx + c (a ≠ 0) is called a quadratic function. By
completing the square, f(x) can be expressed in vertex form as f(x) = a(x − h)2 + k.
•The graph of f is a parabola with vertex (h, k).
•If a > 0, the parabola opens upward, and the vertex is the minimum point. The
minimum value of f is k.
•If a < 0, the parabola opens downward, and the vertex is the maximum point. The
maximum value of f is k.
•The axis of symmetry is x = h. This is the vertical line that passes through the vertex.
3.1 – Quadratic Functions and Application
Vertex Formula to Find the Vertex of a Parabola
For f(x) = ax2 + bx + c (a ≠ 0), the vertex is given by:
Express function in vertex form
Sketch the graph
Open up or down
Axis of symmetry
Identify the vertex coordinates
Max. or min. value of the function
x-intercepts: y = 0
State the Domain and Range
y-intercepts: x = 0
3.1 – Quadratic Functions and Application
Use the vertex formula to find the coordinates of the vertex for:
𝑞 𝑥 = 3𝑥 2 − 36𝑥 + 1
36
−(−36)
=6
=
6
2(3)
𝑞 6 = 3(6)2 −36(6) + 1
𝑞 6 = −107
coordinates of the vertex
(6, −107)
3.1 – Quadratic Functions and Application
𝑞 𝑥 = 2𝑥 2 + 8𝑥 − 1
Express function in vertex form
a
Open up or down
Identify the vertex coordinates
Sketch the graph
Axis of symmetry
a
x-intercepts: y = 0
Max. or min. value of the function
State the Domain and Range
y-intercepts: x = 0
𝑞 𝑥 = 2(𝑥 2 + 4𝑥) − 1
4
= 2 22 = 4
2
𝑞 𝑥 = 2(𝑥 + 2)2 −8 − 1
𝑞 𝑥 = 2(𝑥 2 + 4𝑥 + 4 − 4) − 1
coordinates of the vertex
(−2, −9)
𝑞 𝑥 = 2((𝑥 + 2)2 −4) − 1
vertex form
𝑞 𝑥 = 2(𝑥 + 2)2 −9
3.1 – Quadratic Functions and Application
𝑞 𝑥 = 2𝑥 2 + 8𝑥 − 1
𝑞 𝑥 = 2(𝑥 + 2)2 −9
Express function in vertex form
Open up or down
Identify the vertex coordinates
x-intercepts: y = 0
y-intercepts: x = 0
a
up
a
a
a
𝑥 − 𝑖𝑛𝑡: 𝑦 = 0
−8 ± 82 − 4(2)(−1)
𝑥=
2(2)
𝑥 = −4.121, 0.121
Sketch the graph
Axis of symmetry
𝑥 = −2
Max./min. value of the function 𝑚𝑖𝑛. : −9
State the Domain and Range
𝑦 − 𝑖𝑛𝑡: 𝑥 = 0
𝑞 𝑥 = 𝑦 = −1
3.1 – Quadratic Functions and Application
𝑞 𝑥 = 2𝑥 2 + 8𝑥 − 1
𝑥 = −2
(−4.121, 0)
(0.121, 0)
(0, −1)
(−2, −9)
3.1 – Quadratic Functions and Application
Given a sheet of aluminum that measures 20 inches by 8 inches:
a) Write the equation that represents the volume of a rectangular gutter
that can be formed from the sheet of aluminum.
b) At what value of x does the maximum volume occur?
c) What is the maximum volume?
𝑎)
𝑐)
𝑉=𝑙𝑤ℎ
𝑉 = 20 8 − 2𝑥 𝑥
𝑉(𝑥) =
𝑏)
x
8 − 2𝑥
x
−40𝑥 2
+ 160𝑥
max 𝑣𝑜𝑙𝑢𝑚𝑒
𝑉(𝑥) = −40𝑥 2 + 160𝑥
𝑉(2) = −40 2
2
+ 160(2)
𝑝𝑎𝑟𝑎𝑏𝑜𝑙𝑎, 𝑜𝑝𝑒𝑛𝑠 𝑑𝑜𝑤𝑛
𝑉(2) = 160
∴ 𝑡ℎ𝑒𝑟𝑒 𝑖𝑠 𝑎 𝑚𝑎𝑥𝑖𝑚𝑢𝑚
max 𝑣𝑜𝑙𝑢𝑚𝑒 = 160 𝑖𝑛3
max 𝑜𝑐𝑐𝑢𝑟𝑠 𝑎𝑡 𝑡ℎ𝑒 𝑣𝑒𝑟𝑡𝑒𝑥
−𝑏
−160
𝑥=
=
=2
2𝑎 2(−40)
3.2 – Introduction to Polynomial Functions
Defn:
Polynomial function
In the form of: 𝑓 𝑥 = 𝑎𝑛 𝑥 𝑛 + 𝑎𝑛−1 𝑥 𝑛−1 + ⋯ 𝑎1 𝑥 + 𝑎0 .
The coefficients are real numbers.
The exponents are non-negative integers.
The domain of the function is the set of all real numbers.
Are the following functions polynomials?
𝑓 𝑥 = 5𝑥 + 2𝑥 2 − 6𝑥 3 + 3
yes
ℎ 𝑥 = 2𝑥 3 (4𝑥 5 + 3𝑥)
yes
𝑔 𝑥 = 2𝑥 2 − 4𝑥 + 𝑥 − 2
no
2𝑥 3 + 3
𝑘 𝑥 = 5
4𝑥 + 3𝑥
no
3.2 – Introduction to Polynomial Functions
Defn:
Degree of a Function
The largest degree of the function represents the degree of the
function.
The zero function (all coefficients and the constant are zero) does
not have a degree.
State the degree of the following polynomial functions
𝑓 𝑥 = 5𝑥 + 2𝑥 2 − 6𝑥 3 + 3
𝑔 𝑥 = 2𝑥 5 − 4𝑥 3 + 𝑥 − 2
3
ℎ 𝑥 = 2𝑥 3 (4𝑥 5 + 3𝑥)
8
5
𝑘 𝑥 = 4𝑥 3 + 6𝑥 11 − 𝑥 10 + 𝑥 12
12
3.2 – Introduction to Polynomial Functions
End Behavior of a Function (Leading Term Test)
If 𝑓 𝑥 = 𝑎𝑛 𝑥 𝑛 + 𝑎𝑛−1 𝑥 𝑛−1 + ⋯ 𝑎1 𝑥 + 𝑎0 , then the end behaviors of
the graph will depend on the first term of the function, 𝑎𝑥 𝑛 .
If 𝑓 𝑥 = 𝑎𝑥 𝑛 and n is even, then both ends will approach +.
If 𝑓 𝑥 = −𝑎𝑥 𝑛 and n is even, then both ends will approach –.
If 𝑓 𝑥 = 𝑎𝑥 𝑛 and n is odd,
then as x  – , 𝑓 𝑥  – and as x  , 𝑓 𝑥  .
If 𝑓 𝑥 = −𝑎𝑥 𝑛 and n is odd,
then as x  – , 𝑓 𝑥   and as x  , 𝑓 𝑥  –.
3.2 – Introduction to Polynomial Functions
End Behavior of a Function
𝑓 𝑥 = 𝑎𝑥 𝑛 and n is even
𝑓 𝑥 = −𝑎𝑥 𝑛 and n is even
𝑓 𝑥 = 𝑎𝑥 𝑛 and n is odd
𝑓 𝑥 = −𝑎𝑥 𝑛 and n is odd
3.2 – Introduction to Polynomial Functions
Defn:
Real Zero of a function
If f(r) = 0 and r is a real number, then r is a real zero of the function.
Equivalent Statements for a Real Zero
r is a real zero of the function.
r is an x-intercept of the graph of the function.
x – r is a factor of the function.
r is a solution to the function f(x) = 0
3.2 – Introduction to Polynomial Functions
Defn:
Multiplicity
The number of times a factor (m) of a function is repeated is
referred to its multiplicity (zero multiplicity of m).
Zero Multiplicity of an Even Number
The graph of the function touches the x-axis but does not cross it.
Zero Multiplicity of an Odd Number
The graph of the function crosses the x-axis.
3.2 – Introduction to Polynomial Functions
Identify the zeros and their multiplicity
𝑓 𝑥 = 𝑥−3 𝑥+2
3
3 is a zero with a multiplicity of 1.
Graph crosses the x-axis.
-2 is a zero with a multiplicity of 3.
Graph crosses the x-axis.
𝑔 𝑥 =5 𝑥+4 𝑥−7
2
-4 is a zero with a multiplicity of 1.
Graph crosses the x-axis.
7 is a zero with a multiplicity of 2.
Graph touches the x-axis.
𝑔 𝑥 = 𝑥 + 1 (𝑥 − 4) 𝑥 − 2
2
-1 is a zero with a multiplicity of 1.
Graph crosses the x-axis.
4 is a zero with a multiplicity of 1.
Graph crosses the x-axis.
2 is a zero with a multiplicity of 2.
Graph touches the x-axis.
3.2 – Introduction to Polynomial Functions
Turning Points
The point where a function changes directions from increasing to
decreasing or from decreasing to increasing.
If a function has a degree of n, then it has at most n – 1 turning points.
If the graph of a polynomial function has t number of turning points,
then the function has at least a degree of t + 1 .
What is the most number of turning points the following polynomial
functions could have?
𝑓 𝑥 = 5𝑥 + 2𝑥 2 − 6𝑥 3 + 3
3-1
2
ℎ 𝑥 = 2𝑥 3 (4𝑥 5 + 3𝑥)
8-1
7
𝑔 𝑥 = 2𝑥 5 − 4𝑥 3 + 𝑥 − 2
5-1
4
𝑘 𝑥 = 4𝑥 3 + 6𝑥 11 − 𝑥 10 + 𝑥 12
12-1
11
3.2 – Introduction to Polynomial Functions
Intermediate Value Theorem
In a polynomial function, if a < b and f(a) and f(b) are of opposite
signs, then there is at least one real zero between a and b.
(𝑏, 𝑓 𝑏 )
(𝑎, 𝑓 𝑎 )
𝑟𝑒𝑎𝑙 𝑧𝑒𝑟𝑜
𝑟𝑒𝑎𝑙 𝑧𝑒𝑟𝑜
(𝑎, 𝑓 𝑎 )
(𝑏, 𝑓 𝑏 )
3.2 – Introduction to Polynomial Functions
Intermediate Value Theorem
Do the following polynomial functions have at least one real zero in the
given interval?
𝑓 𝑥 = 2𝑥 3 − 3𝑥 2 − 2
𝑓 𝑥 = 2𝑥 3 − 3𝑥 2 − 2
[0, 2]
[3, 6]
𝑓 0 = −2
𝑓 2 = 2
𝑦𝑒𝑠
𝑛𝑜𝑡 𝑒𝑛𝑜𝑢𝑔ℎ 𝑖𝑛𝑓𝑜𝑟𝑚𝑎𝑡𝑖𝑜𝑛
𝑓 𝑥 = 𝑥 4 − 2𝑥 2 − 3𝑥 − 3
[−5, −2]
𝑓 −5 = 587
𝑓 6 = 322
𝑓 3 = 25
𝑓 −2 = 11
𝑛𝑜𝑡 𝑒𝑛𝑜𝑢𝑔ℎ 𝑖𝑛𝑓𝑜𝑟𝑚𝑎𝑡𝑖𝑜𝑛
𝑓 𝑥 = 𝑥 4 − 2𝑥 2 − 3𝑥 − 3
[−1, 3]
𝑓 −1 = −1
𝑓 3 = 51
𝑦𝑒𝑠
3.2 – Introduction to Polynomial Functions
Graph a possible function.
𝑔 𝑥 = 𝑥 + 1 (𝑥 − 4) 𝑥 − 2 2
End Behavior
Positive coefficient
w/even power
Left up; Right up
Zeros and Multiplicity
𝑥 = −1, zero mult. of 1
𝑐𝑟𝑜𝑠𝑠𝑒𝑠 𝑎𝑡 𝑥 = −1
𝑥 = 4, zero mult. of 1
𝑐𝑟𝑜𝑠𝑠𝑒𝑠 𝑎𝑡 𝑥 = 4
𝑥 = 2, zero mult. of 2
𝑡𝑜𝑢𝑐ℎ𝑒𝑠 𝑎𝑡 𝑥 = 2
y-int. → 𝑥 = 0
𝑔 𝑥 = 𝑦 = 0 + 1 (0 − 4) 0 − 2
𝑦 = −16
Turning points
Degree of function: 4
4−1=3
𝐴𝑡 𝑚𝑜𝑠𝑡, 3 𝑡𝑢𝑟𝑛𝑖𝑛𝑔 𝑝𝑜𝑖𝑛𝑡𝑠
2
3.2 – Introduction to Polynomial Functions
Graph a possible function
𝑔 𝑥 = 𝑥 + 1 (𝑥 − 4) 𝑥 − 2
-1
(0, -16)
2
4
2
3.1 – Quadratic Functions and Application
𝑞 𝑥 = −2 𝑥 + 3
2
+8