sec 3.1 and 3 - university of minnesotahankx003/fall2012/lectures/ch3sec1an… · sec 3.1 and 3.3...
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Sec 3.1 and 3.3
Linear and Quadratic Functions
Math 1051 - Precalculus I
Linear and Quadratic Functions Sec 3.1 and 3.3
Sec 3.1 and 3.3 Linear and Quadratic Functions
Is f (x) = x2−x3x−2 even, odd, or neither?
Ans: Neither even nor odd
Linear and Quadratic Functions Sec 3.1 and 3.3
Sec 3.1 and 3.3 Linear and Quadratic Functions
Is f (x) = x2−x3x−2 even, odd, or neither?
Ans: Neither even nor odd
Linear and Quadratic Functions Sec 3.1 and 3.3
Check exam key on my web site
You should look at what you missed on the exam and figure outwhy you did not know how to do the problem given what we didin review.
Did you attend the review?Were you paying attention at the review?Did you study the review problems enough?Did you study old exam questions on my web site?Did you study the checkpoints thoroughly?How will you study differently for Exam 3?
Learn from your mistakes so you don’t repeat them!
Linear and Quadratic Functions Sec 3.1 and 3.3
Check exam key on my web site
You should look at what you missed on the exam and figure outwhy you did not know how to do the problem given what we didin review.
Did you attend the review?Were you paying attention at the review?Did you study the review problems enough?Did you study old exam questions on my web site?Did you study the checkpoints thoroughly?How will you study differently for Exam 3?
Learn from your mistakes so you don’t repeat them!
Linear and Quadratic Functions Sec 3.1 and 3.3
Check exam key on my web site
You should look at what you missed on the exam and figure outwhy you did not know how to do the problem given what we didin review.
Did you attend the review?
Were you paying attention at the review?Did you study the review problems enough?Did you study old exam questions on my web site?Did you study the checkpoints thoroughly?How will you study differently for Exam 3?
Learn from your mistakes so you don’t repeat them!
Linear and Quadratic Functions Sec 3.1 and 3.3
Check exam key on my web site
You should look at what you missed on the exam and figure outwhy you did not know how to do the problem given what we didin review.
Did you attend the review?Were you paying attention at the review?
Did you study the review problems enough?Did you study old exam questions on my web site?Did you study the checkpoints thoroughly?How will you study differently for Exam 3?
Learn from your mistakes so you don’t repeat them!
Linear and Quadratic Functions Sec 3.1 and 3.3
Check exam key on my web site
You should look at what you missed on the exam and figure outwhy you did not know how to do the problem given what we didin review.
Did you attend the review?Were you paying attention at the review?Did you study the review problems enough?
Did you study old exam questions on my web site?Did you study the checkpoints thoroughly?How will you study differently for Exam 3?
Learn from your mistakes so you don’t repeat them!
Linear and Quadratic Functions Sec 3.1 and 3.3
Check exam key on my web site
You should look at what you missed on the exam and figure outwhy you did not know how to do the problem given what we didin review.
Did you attend the review?Were you paying attention at the review?Did you study the review problems enough?Did you study old exam questions on my web site?
Did you study the checkpoints thoroughly?How will you study differently for Exam 3?
Learn from your mistakes so you don’t repeat them!
Linear and Quadratic Functions Sec 3.1 and 3.3
Check exam key on my web site
You should look at what you missed on the exam and figure outwhy you did not know how to do the problem given what we didin review.
Did you attend the review?Were you paying attention at the review?Did you study the review problems enough?Did you study old exam questions on my web site?Did you study the checkpoints thoroughly?
How will you study differently for Exam 3?
Learn from your mistakes so you don’t repeat them!
Linear and Quadratic Functions Sec 3.1 and 3.3
Check exam key on my web site
You should look at what you missed on the exam and figure outwhy you did not know how to do the problem given what we didin review.
Did you attend the review?Were you paying attention at the review?Did you study the review problems enough?Did you study old exam questions on my web site?Did you study the checkpoints thoroughly?How will you study differently for Exam 3?
Learn from your mistakes so you don’t repeat them!
Linear and Quadratic Functions Sec 3.1 and 3.3
Check exam key on my web site
You should look at what you missed on the exam and figure outwhy you did not know how to do the problem given what we didin review.
Did you attend the review?Were you paying attention at the review?Did you study the review problems enough?Did you study old exam questions on my web site?Did you study the checkpoints thoroughly?How will you study differently for Exam 3?
Learn from your mistakes so you don’t repeat them!
Linear and Quadratic Functions Sec 3.1 and 3.3
Check exam key on my web site
You should look at what you missed on the exam and figure outwhy you did not know how to do the problem given what we didin review.
Did you attend the review?Were you paying attention at the review?Did you study the review problems enough?Did you study old exam questions on my web site?Did you study the checkpoints thoroughly?How will you study differently for Exam 3?
Learn from your mistakes so you don’t repeat them!
Linear and Quadratic Functions Sec 3.1 and 3.3
Linear Functions
f (x) = mx + b
Domain:
All real numbers
Range:
All real numbers
y -intercept:
(0,b)
x-intercept:
(− b
m ,0)
Where is the function increasing:
When m > 0, (−∞,∞)
Where is the function decreasing:
When m < 0, (−∞,∞)
Where is the function constant:
When m = 0, (−∞,∞)
Maxima and Minima:
None
Linear and Quadratic Functions Sec 3.1 and 3.3
Linear Functions
f (x) = mx + b
Domain:
All real numbers
Range:
All real numbers
y -intercept:
(0,b)
x-intercept:
(− b
m ,0)
Where is the function increasing:
When m > 0, (−∞,∞)
Where is the function decreasing:
When m < 0, (−∞,∞)
Where is the function constant:
When m = 0, (−∞,∞)
Maxima and Minima:
None
Linear and Quadratic Functions Sec 3.1 and 3.3
Linear Functions
f (x) = mx + b
Domain: All real numbersRange:
All real numbers
y -intercept:
(0,b)
x-intercept:
(− b
m ,0)
Where is the function increasing:
When m > 0, (−∞,∞)
Where is the function decreasing:
When m < 0, (−∞,∞)
Where is the function constant:
When m = 0, (−∞,∞)
Maxima and Minima:
None
Linear and Quadratic Functions Sec 3.1 and 3.3
Linear Functions
f (x) = mx + b
Domain: All real numbersRange: All real numbersy -intercept:
(0,b)
x-intercept:
(− b
m ,0)
Where is the function increasing:
When m > 0, (−∞,∞)
Where is the function decreasing:
When m < 0, (−∞,∞)
Where is the function constant:
When m = 0, (−∞,∞)
Maxima and Minima:
None
Linear and Quadratic Functions Sec 3.1 and 3.3
Linear Functions
f (x) = mx + b
Domain: All real numbersRange: All real numbersy -intercept: (0,b)x-intercept:
(− b
m ,0)
Where is the function increasing:
When m > 0, (−∞,∞)
Where is the function decreasing:
When m < 0, (−∞,∞)
Where is the function constant:
When m = 0, (−∞,∞)
Maxima and Minima:
None
Linear and Quadratic Functions Sec 3.1 and 3.3
Linear Functions
f (x) = mx + b
Domain: All real numbersRange: All real numbersy -intercept: (0,b)x-intercept:
(− b
m ,0)
Where is the function increasing:
When m > 0, (−∞,∞)
Where is the function decreasing:
When m < 0, (−∞,∞)
Where is the function constant:
When m = 0, (−∞,∞)
Maxima and Minima:
None
Linear and Quadratic Functions Sec 3.1 and 3.3
Linear Functions
f (x) = mx + b
Domain: All real numbersRange: All real numbersy -intercept: (0,b)x-intercept:
(− b
m ,0)
Where is the function increasing: When m > 0, (−∞,∞)
Where is the function decreasing:
When m < 0, (−∞,∞)
Where is the function constant:
When m = 0, (−∞,∞)
Maxima and Minima:
None
Linear and Quadratic Functions Sec 3.1 and 3.3
Linear Functions
f (x) = mx + b
Domain: All real numbersRange: All real numbersy -intercept: (0,b)x-intercept:
(− b
m ,0)
Where is the function increasing: When m > 0, (−∞,∞)
Where is the function decreasing: When m < 0, (−∞,∞)
Where is the function constant:
When m = 0, (−∞,∞)
Maxima and Minima:
None
Linear and Quadratic Functions Sec 3.1 and 3.3
Linear Functions
f (x) = mx + b
Domain: All real numbersRange: All real numbersy -intercept: (0,b)x-intercept:
(− b
m ,0)
Where is the function increasing: When m > 0, (−∞,∞)
Where is the function decreasing: When m < 0, (−∞,∞)
Where is the function constant: When m = 0, (−∞,∞)
Maxima and Minima:
None
Linear and Quadratic Functions Sec 3.1 and 3.3
Linear Functions
f (x) = mx + b
Domain: All real numbersRange: All real numbersy -intercept: (0,b)x-intercept:
(− b
m ,0)
Where is the function increasing: When m > 0, (−∞,∞)
Where is the function decreasing: When m < 0, (−∞,∞)
Where is the function constant: When m = 0, (−∞,∞)
Maxima and Minima: None
Linear and Quadratic Functions Sec 3.1 and 3.3
Average Rate of Change for f (x) = mx + b
ARC =4y4x
=f (c)− f (a)
c − a
ARC = m
Linear and Quadratic Functions Sec 3.1 and 3.3
Average Rate of Change for f (x) = mx + b
ARC =4y4x
=f (c)− f (a)
c − a
ARC = m
Linear and Quadratic Functions Sec 3.1 and 3.3
Average Rate of Change for f (x) = mx + b
ARC =4y4x
=f (c)− f (a)
c − a
ARC = m
Linear and Quadratic Functions Sec 3.1 and 3.3
Inequalities with linear functions and graphs
Linear and Quadratic Functions Sec 3.1 and 3.3
Supply and Demand
Suppose we observe hot dog sales at a baseball game
Supply: S(p) = −2000 + 3000p
Demand: D(p) = 10,000− 1000p
Linear and Quadratic Functions Sec 3.1 and 3.3
Good questions:
What is the equilibrium in price and quantity?When is demand lower than supply?
Linear and Quadratic Functions Sec 3.1 and 3.3
Good questions:What is the equilibrium in price and quantity?
When is demand lower than supply?
Linear and Quadratic Functions Sec 3.1 and 3.3
Good questions:What is the equilibrium in price and quantity?When is demand lower than supply?
Linear and Quadratic Functions Sec 3.1 and 3.3
When is demand lower than supply?
5 10
-5000
5000
10 000
Linear and Quadratic Functions Sec 3.1 and 3.3
Quadratic Functions
f (x) = ax2 + bx + c
Domain:
All real numbers
Range:
Depends on sign of a and location of vertex
y -intercept:
(0, c)
x-intercept:
(−b ±
√b2 − 4ac
2a,0
)
Increasing:
Depends on a, b, c
Decreasing:
Depends on a, b, c
Constant:
Never
Maxima and Minima:
At the vertex
Linear and Quadratic Functions Sec 3.1 and 3.3
Quadratic Functions
f (x) = ax2 + bx + c
Domain:
All real numbers
Range:
Depends on sign of a and location of vertex
y -intercept:
(0, c)
x-intercept:
(−b ±
√b2 − 4ac
2a,0
)
Increasing:
Depends on a, b, c
Decreasing:
Depends on a, b, c
Constant:
Never
Maxima and Minima:
At the vertex
Linear and Quadratic Functions Sec 3.1 and 3.3
Quadratic Functions
f (x) = ax2 + bx + c
Domain: All real numbersRange:
Depends on sign of a and location of vertex
y -intercept:
(0, c)
x-intercept:
(−b ±
√b2 − 4ac
2a,0
)
Increasing:
Depends on a, b, c
Decreasing:
Depends on a, b, c
Constant:
Never
Maxima and Minima:
At the vertex
Linear and Quadratic Functions Sec 3.1 and 3.3
Quadratic Functions
f (x) = ax2 + bx + c
Domain: All real numbersRange: Depends on sign of a and location of vertexy -intercept:
(0, c)
x-intercept:
(−b ±
√b2 − 4ac
2a,0
)
Increasing:
Depends on a, b, c
Decreasing:
Depends on a, b, c
Constant:
Never
Maxima and Minima:
At the vertex
Linear and Quadratic Functions Sec 3.1 and 3.3
Quadratic Functions
f (x) = ax2 + bx + c
Domain: All real numbersRange: Depends on sign of a and location of vertexy -intercept: (0, c)x-intercept:
(−b ±
√b2 − 4ac
2a,0
)
Increasing:
Depends on a, b, c
Decreasing:
Depends on a, b, c
Constant:
Never
Maxima and Minima:
At the vertex
Linear and Quadratic Functions Sec 3.1 and 3.3
Quadratic Functions
f (x) = ax2 + bx + c
Domain: All real numbersRange: Depends on sign of a and location of vertexy -intercept: (0, c)x-intercept: (
−b ±√
b2 − 4ac2a
,0
)
Increasing:
Depends on a, b, c
Decreasing:
Depends on a, b, c
Constant:
Never
Maxima and Minima:
At the vertex
Linear and Quadratic Functions Sec 3.1 and 3.3
Quadratic Functions
f (x) = ax2 + bx + c
Domain: All real numbersRange: Depends on sign of a and location of vertexy -intercept: (0, c)x-intercept: (
−b ±√
b2 − 4ac2a
,0
)
Increasing: Depends on a, b, cDecreasing:
Depends on a, b, c
Constant:
Never
Maxima and Minima:
At the vertex
Linear and Quadratic Functions Sec 3.1 and 3.3
Quadratic Functions
f (x) = ax2 + bx + c
Domain: All real numbersRange: Depends on sign of a and location of vertexy -intercept: (0, c)x-intercept: (
−b ±√
b2 − 4ac2a
,0
)
Increasing: Depends on a, b, cDecreasing: Depends on a, b, cConstant:
Never
Maxima and Minima:
At the vertex
Linear and Quadratic Functions Sec 3.1 and 3.3
Quadratic Functions
f (x) = ax2 + bx + c
Domain: All real numbersRange: Depends on sign of a and location of vertexy -intercept: (0, c)x-intercept: (
−b ±√
b2 − 4ac2a
,0
)
Increasing: Depends on a, b, cDecreasing: Depends on a, b, cConstant: NeverMaxima and Minima:
At the vertex
Linear and Quadratic Functions Sec 3.1 and 3.3
Quadratic Functions
f (x) = ax2 + bx + c
Domain: All real numbersRange: Depends on sign of a and location of vertexy -intercept: (0, c)x-intercept: (
−b ±√
b2 − 4ac2a
,0
)
Increasing: Depends on a, b, cDecreasing: Depends on a, b, cConstant: NeverMaxima and Minima: At the vertex
Linear and Quadratic Functions Sec 3.1 and 3.3
Some good information about quadratic functions
Linear and Quadratic Functions Sec 3.1 and 3.3
If we start with the standard form of a parabola
f (x) = ax2 + bx + c
we can “complete the square” to get a new form.
Why did I do that???
Linear and Quadratic Functions Sec 3.1 and 3.3
If we start with the standard form of a parabola
f (x) = ax2 + bx + c
we can “complete the square” to get a new form.
Why did I do that???
Linear and Quadratic Functions Sec 3.1 and 3.3
f (x) = a(
x +b2a
)2
+4ac − b2
4a
= a(x − h)2 + k
If h = − b2a and k = 4ac−b2
4a
This is called the vertex form of a quadratic equation.
Linear and Quadratic Functions Sec 3.1 and 3.3
f (x) = a(
x +b2a
)2
+4ac − b2
4a
= a(x − h)2 + k
If h = − b2a and k = 4ac−b2
4a
This is called the vertex form of a quadratic equation.
Linear and Quadratic Functions Sec 3.1 and 3.3
Vertex Form
f (x) = a(x − h)2 + k
h gives the horizontal shift, so it’s the x-coordinate of thevertex.k gives the vertical shift, so it’s the y -coordinate of thevertex.a gives the stretch or compression.If a is negative reflect about the x-axis.
Linear and Quadratic Functions Sec 3.1 and 3.3
Vertex Form
f (x) = a(x − h)2 + k
h gives the horizontal shift, so it’s the x-coordinate of thevertex.
k gives the vertical shift, so it’s the y -coordinate of thevertex.a gives the stretch or compression.If a is negative reflect about the x-axis.
Linear and Quadratic Functions Sec 3.1 and 3.3
Vertex Form
f (x) = a(x − h)2 + k
h gives the horizontal shift, so it’s the x-coordinate of thevertex.k gives the vertical shift, so it’s the y -coordinate of thevertex.
a gives the stretch or compression.If a is negative reflect about the x-axis.
Linear and Quadratic Functions Sec 3.1 and 3.3
Vertex Form
f (x) = a(x − h)2 + k
h gives the horizontal shift, so it’s the x-coordinate of thevertex.k gives the vertical shift, so it’s the y -coordinate of thevertex.a gives the stretch or compression.
If a is negative reflect about the x-axis.
Linear and Quadratic Functions Sec 3.1 and 3.3
Vertex Form
f (x) = a(x − h)2 + k
h gives the horizontal shift, so it’s the x-coordinate of thevertex.k gives the vertical shift, so it’s the y -coordinate of thevertex.a gives the stretch or compression.If a is negative reflect about the x-axis.
Linear and Quadratic Functions Sec 3.1 and 3.3
Graph using transformations
f (x) = −2x2 + 6x + 2
-2 2 4 6
-4
-2
2
4
6
Linear and Quadratic Functions Sec 3.1 and 3.3
Graph using transformations
f (x) = −2x2 + 6x + 2
-2 2 4 6
-4
-2
2
4
6
Linear and Quadratic Functions Sec 3.1 and 3.3
Or, you can make the graph quickly this way...
Linear and Quadratic Functions Sec 3.1 and 3.3
Given the graph below, find the corresponding function
-2 2 4 6
-2
2
4
6
Linear and Quadratic Functions Sec 3.1 and 3.3
Read section 3.4 for Friday.
Linear and Quadratic Functions Sec 3.1 and 3.3