•The axis of symmetry is x = h. This is the vertical line that passes through the vertex.

3.1 – Quadratic Functions and Application

Quadratic FunctionsA 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.

Vertex Formula to Find the Vertex of a ParabolaFor f(x) = ax2 + bx + c (a ≠ 0), the vertex is given by:

3.1 – Quadratic Functions and Application

Quadratic Functions

Analyzing Quadratic Functions

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

Quadratic Functions

𝑞 (𝑥 )=3 𝑥2−36𝑥+1

Use the vertex formula to find the coordinates of the vertex for:

−(−36)2(3)

=¿366

=¿6

𝑞 (6 )=3 (6)2−36 (6)+1𝑞 (6 )=−107

coordinates of the vertex

(6 ,−107)

3.1 – Quadratic Functions and Application

Quadratic Functions

Analyzing Quadratic Functions

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

𝑞 (𝑥 )=2 𝑥2+8𝑥−1

𝑞 (𝑥 )=2(𝑥2+4 𝑥)−142=2 22=4

𝑞 (𝑥 )=2(𝑥2+4 𝑥+4−4 )−1𝑞 (𝑥 )=2((𝑥+2)2−4)−1

𝑞 (𝑥 )=2(𝑥+2)2−8−1

𝑞 (𝑥 )=2(𝑥+2)2−9coordinates of the vertex

(−2 ,−9)

vertex form

a

a

3.1 – Quadratic Functions and Application

Quadratic Functions

Analyzing Quadratic Functions

Express function in vertex form Sketch the graph

Open up or down Axis of symmetry

Identify the vertex coordinates Max./min. value of the function

x-intercepts: y = 0 State the Domain and Range

y-intercepts: x = 0

𝑞 (𝑥 )=2 𝑥2+8𝑥−1

𝑥−𝑖𝑛𝑡 :𝑦=0

𝑥=−8±√82−4 (2)(−1)

2(2)

𝑥=−4.121 ,0.121

𝑞 (𝑥 )=𝑦=−1

𝑚𝑖𝑛 . :−9𝑥=−2

a

aup

𝑦−𝑖𝑛𝑡 :𝑥=0

aa

𝑞 (𝑥 )=2(𝑥+2)2−9

(−4.121 ,0) (0.121 ,0)(0 ,−1)

(−2 ,−9)

𝑞 (𝑥 )=2 𝑥2+8𝑥−1

3.1 – Quadratic Functions and Application

Quadratic Functions

𝑥=−2

8 in.

20 in.

3.1 – Quadratic Functions and Application

Quadratic Functions

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?

20 in.

xx

8−2𝑥

𝑉=𝑙𝑤 h𝑉=20 (8−2 𝑥 ) 𝑥𝑉 (𝑥)=−40𝑥2+160 𝑥

𝑎 ¿

𝑏¿

𝑝𝑎𝑟𝑎𝑏𝑜𝑙𝑎 ,𝑜𝑝𝑒𝑛𝑠 𝑑𝑜𝑤𝑛∴ h𝑡 𝑒𝑟𝑒 𝑖𝑠𝑎𝑚𝑎𝑥𝑖𝑚𝑢𝑚max𝑜𝑐𝑐𝑢𝑟𝑠𝑎𝑡 h𝑡 𝑒𝑣𝑒𝑟𝑡𝑒𝑥

𝑥=−𝑏2𝑎

=−160

2(−40)=2

𝑐 ¿𝑉 (𝑥)=−40𝑥2+160 𝑥𝑉 (2)=−40 (2 )2+160(2)𝑉 (2)=160

max 𝑣𝑜𝑙𝑢𝑚𝑒

max 𝑣𝑜𝑙𝑢𝑚𝑒=160 𝑖𝑛3

Defn: Polynomial function In the form of: . The coefficients are real numbers. The exponents are non-negative integers. The domain of the function is the set of all real numbers.

𝑓 (𝑥 )=5𝑥+2 𝑥2−6 𝑥3+3 𝑔 (𝑥 )=2 𝑥2−4 𝑥+√𝑥−2

h (𝑥 )=2 𝑥3 (4 𝑥5+3 𝑥)

Are the following functions polynomials?

yes no

yes

𝑘 (𝑥 )= 2𝑥3+34 𝑥5+3𝑥 no

3.2 – Introduction to Polynomial Functions

Defn:

Degree of a FunctionThe 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.

𝑓 (𝑥 )=5𝑥+2 𝑥2−6 𝑥3+3 𝑔 (𝑥 )=2 𝑥5−4 𝑥3+𝑥−2

h (𝑥 )=2 𝑥3 (4 𝑥5+3 𝑥)

3 5

8

𝑘 (𝑥 )=4 𝑥3+6 𝑥11−𝑥10+𝑥12

12

State the degree of the following polynomial functions

3.2 – Introduction to Polynomial Functions

If and n is even, then both ends will approach +.

End Behavior of a Function (Leading Term Test)If , then the end behaviors of the graph will depend on the first term of the function, .

If and n is even, then both ends will approach –.

If and n is odd, then as x – , – and as x , .

If and n is odd, then as x – , and as x , –.

3.2 – Introduction to Polynomial Functions

and n is even

End Behavior of a Function

and n is even

and n is odd and n is odd

3.2 – Introduction to Polynomial Functions

Defn:

Real Zero of a function

r is a real zero of the function.

r is an x-intercept of the graph of the function.

Equivalent Statements for a Real Zero

x – r is a factor of the function.

r is a solution to the function f(x) = 0

If f(r) = 0 and r is a real number, then r is a real zero of the function.

3.2 – Introduction to Polynomial Functions

Defn:

The graph of the function touches the x-axis but does not cross it. Zero Multiplicity of an Even Number

MultiplicityThe number of times a factor (m) of a function is repeated is referred to its multiplicity (zero multiplicity of m).

The graph of the function crosses the x-axis. Zero Multiplicity of an Odd Number

3.2 – Introduction to Polynomial Functions

3 is a zero with a multiplicity of

𝑓 (𝑥 )=(𝑥−3 ) (𝑥+2 )3Identify the zeros and their multiplicity

3.-2 is a zero with a multiplicity of

1. Graph crosses the x-axis.

Graph crosses the x-axis.

-4 is a zero with a multiplicity of

𝑔 (𝑥 )=5 (𝑥+4 ) (𝑥−7 )2

2.7 is a zero with a multiplicity of

1. Graph crosses the x-axis.

Graph touches the x-axis.

-1 is a zero with a multiplicity of

𝑔 (𝑥 )=(𝑥+1 )(𝑥−4) (𝑥−2 )2

1.4 is a zero with a multiplicity of

1. Graph crosses the x-axis.

Graph crosses the x-axis.

2.2 is a zero with a multiplicity of Graph touches the x-axis.

3.2 – Introduction to Polynomial Functions

If a function has a degree of n, then it has at most n – 1 turning points.

Turning Points

The point where a function changes directions from increasing to decreasing or from decreasing to increasing.

If the graph of a polynomial function has t number of turning points, then the function has at least a degree of t + 1 .

𝑓 (𝑥 )=5𝑥+2 𝑥2−6 𝑥3+3 𝑔 (𝑥 )=2 𝑥5−4 𝑥3+𝑥−2

h (𝑥 )=2 𝑥3 (4 𝑥5+3 𝑥)

3-1 5-1

8-1

𝑘 (𝑥 )=4 𝑥3+6 𝑥11−𝑥10+𝑥12

12-1

What is the most number of turning points the following polynomial functions could have?

2 4

7 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.

(𝑎 , 𝑓 (𝑎 ))

(𝑏 , 𝑓 (𝑏)) (𝑎 , 𝑓 (𝑎 ))

(𝑏 , 𝑓 (𝑏))𝑟𝑒𝑎𝑙 𝑧𝑒𝑟𝑜

𝑟𝑒𝑎𝑙 𝑧𝑒𝑟𝑜

3.2 – Introduction to Polynomial Functions

𝑓 (𝑥 )=2𝑥3−3 𝑥2−2

𝑓 (0 )=¿

Intermediate Value TheoremDo the following polynomial functions have at least one real zero in the given interval?

−2 𝑓 (2 )=¿2𝑦𝑒𝑠

[0 ,2]𝑓 (𝑥 )=2𝑥3−3 𝑥2−2

𝑓 (3 )=¿25 𝑓 (6 )=¿322𝑛𝑜𝑡 h𝑒𝑛𝑜𝑢𝑔 𝑖𝑛𝑓𝑜𝑟𝑚𝑎𝑡𝑖𝑜𝑛

[3 ,6]

𝑓 (𝑥 )=𝑥4−2𝑥2−3 𝑥−3

𝑓 (−5 )=¿587 𝑓 (−2 )=¿11𝑛𝑜𝑡 h𝑒𝑛𝑜𝑢𝑔 𝑖𝑛𝑓𝑜𝑟𝑚𝑎𝑡𝑖𝑜𝑛

[−5 ,−2]𝑓 (𝑥 )=𝑥4−2𝑥2−3 𝑥−3

𝑓 (−1 )=¿−1 𝑓 (3 )=¿51𝑦𝑒𝑠

[−1 ,3]

3.2 – Introduction to Polynomial Functions

Graph a possible function.

, zero mult. of 1

Leading term:

𝑔 (𝑥 )=(𝑥+1 )(𝑥−4) (𝑥−2 )2

𝑥4

Positive coefficient w/even power

End Behavior

3.2 – Introduction to Polynomial Functions

Left up; Right up

Zeros and Multiplicity

, zero mult. of 1

, zero mult. of 2

𝑐𝑟𝑜𝑠𝑠𝑒𝑠 𝑎𝑡 𝑥=−1

𝑐𝑟𝑜𝑠𝑠𝑒𝑠 𝑎𝑡 𝑥=4

h𝑡𝑜𝑢𝑐 𝑒𝑠𝑎𝑡 𝑥=2

y-int.

𝑔 (𝑥 )=𝑦= (0+1 )(0−4) (0−2 )2

𝑦=−16

Turning points

Degree of function: 4

4−1=3𝐴𝑡𝑚𝑜𝑠𝑡 ,3 𝑡𝑢𝑟𝑛𝑖𝑛𝑔𝑝𝑜𝑖𝑛𝑡𝑠

Graph a possible function

𝑔 (𝑥 )=(𝑥+1 )(𝑥−4) (𝑥−2 )2

42-1

3.2 – Introduction to Polynomial Functions

(0, -16)

3.1 – Quadratic Functions and Application

Quadratic Functions

Analyzing Quadratic Functions

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

𝑞 (𝑥 )=−2 (𝑥+3 )2+8