Sec 3.1 and 3 - University of Minnesotahankx003/Fall2012/Lectures/Ch3Sec1an… · Sec 3.1 and 3.3...

transcript

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

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

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!

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?

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?

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?

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?

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?

Linear Functions

f (x) = mx + b

Domain:

All real numbers

Range:

All real numbers

y -intercept:

x-intercept:

(− b

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:

Linear Functions

f (x) = mx + b

Domain:

All real numbers

Range:

All real numbers

y -intercept:

x-intercept:

(− b

When m > 0, (−∞,∞)

When m < 0, (−∞,∞)

When m = 0, (−∞,∞)

Maxima and Minima:

Linear Functions

f (x) = mx + b

Domain: All real numbersRange:

All real numbers

y -intercept:

x-intercept:

(− b

When m > 0, (−∞,∞)

When m < 0, (−∞,∞)

When m = 0, (−∞,∞)

Maxima and Minima:

Linear Functions

f (x) = mx + b

Domain: All real numbersRange: All real numbersy -intercept:

x-intercept:

(− b

When m > 0, (−∞,∞)

When m < 0, (−∞,∞)

When m = 0, (−∞,∞)

Maxima and Minima:

Linear Functions

f (x) = mx + b

Domain: All real numbersRange: All real numbersy -intercept: (0,b)x-intercept:

(− b

When m > 0, (−∞,∞)

When m < 0, (−∞,∞)

When m = 0, (−∞,∞)

Maxima and Minima:

Linear Functions

f (x) = mx + b

(− b

When m > 0, (−∞,∞)

When m < 0, (−∞,∞)

When m = 0, (−∞,∞)

Maxima and Minima:

Linear Functions

f (x) = mx + b

(− b

Where is the function increasing: When m > 0, (−∞,∞)

When m < 0, (−∞,∞)

When m = 0, (−∞,∞)

Maxima and Minima:

Linear Functions

f (x) = mx + b

(− b

Where is the function decreasing: When m < 0, (−∞,∞)

When m = 0, (−∞,∞)

Maxima and Minima:

Linear Functions

f (x) = mx + b

(− b

Where is the function constant: When m = 0, (−∞,∞)

Maxima and Minima:

Linear Functions

f (x) = mx + b

(− b

Where is the function constant: When m = 0, (−∞,∞)

Maxima and Minima: None

Average Rate of Change for f (x) = mx + b

ARC =4y4x

=f (c)− f (a)

c − a

ARC = m

ARC =4y4x

=f (c)− f (a)

c − a

ARC = m

ARC =4y4x

=f (c)− f (a)

c − a

ARC = m

Inequalities with linear functions and graphs

Supply and Demand

Suppose we observe hot dog sales at a baseball game

Supply: S(p) = −2000 + 3000p

Demand: D(p) = 10,000− 1000p

Good questions:

What is the equilibrium in price and quantity?When is demand lower than supply?

Good questions:What is the equilibrium in price and quantity?

When is demand lower than supply?

Good questions:What is the equilibrium in price and quantity?When is demand lower than supply?

When is demand lower than supply?

10 000

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

Increasing:

Depends on a, b, c

Decreasing:

Depends on a, b, c

Constant:

Maxima and Minima:

At the vertex

Quadratic Functions

f (x) = ax2 + bx + c

Domain:

All real numbers

Range:

y -intercept:

(0, c)

x-intercept:

(−b ±

√b2 − 4ac

Increasing:

Depends on a, b, c

Decreasing:

Depends on a, b, c

Constant:

Maxima and Minima:

At the vertex

Quadratic Functions

f (x) = ax2 + bx + c

Domain: All real numbersRange:

y -intercept:

(0, c)

x-intercept:

(−b ±

√b2 − 4ac

Increasing:

Depends on a, b, c

Decreasing:

Depends on a, b, c

Constant:

Maxima and Minima:

At the vertex

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

Increasing:

Depends on a, b, c

Decreasing:

Depends on a, b, c

Constant:

Maxima and Minima:

At the vertex

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

Increasing:

Depends on a, b, c

Decreasing:

Depends on a, b, c

Constant:

Maxima and Minima:

At the vertex

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

Increasing:

Depends on a, b, c

Decreasing:

Depends on a, b, c

Constant:

Maxima and Minima:

At the vertex

Quadratic Functions

f (x) = ax2 + bx + c

−b ±√

b2 − 4ac2a

Increasing: Depends on a, b, cDecreasing:

Depends on a, b, c

Constant:

Maxima and Minima:

At the vertex

Quadratic Functions

f (x) = ax2 + bx + c

−b ±√

b2 − 4ac2a

Increasing: Depends on a, b, cDecreasing: Depends on a, b, cConstant:

Maxima and Minima:

At the vertex

Quadratic Functions

f (x) = ax2 + bx + c

−b ±√

b2 − 4ac2a

Increasing: Depends on a, b, cDecreasing: Depends on a, b, cConstant: NeverMaxima and Minima:

At the vertex

Quadratic Functions

f (x) = ax2 + bx + c

−b ±√

b2 − 4ac2a

Increasing: Depends on a, b, cDecreasing: Depends on a, b, cConstant: NeverMaxima and Minima: At the vertex

Some good information about quadratic functions

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???

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???

f (x) = a(

x +b2a

+4ac − b2

= a(x − h)2 + k

If h = − b2a and k = 4ac−b2

This is called the vertex form of a quadratic equation.

f (x) = a(

x +b2a

+4ac − b2

= a(x − h)2 + k

If h = − b2a and k = 4ac−b2

This is called the vertex form of a quadratic equation.

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.

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.

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.

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.

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.

Graph using transformations

f (x) = −2x2 + 6x + 2

-2 2 4 6

Graph using transformations

f (x) = −2x2 + 6x + 2

-2 2 4 6

Or, you can make the graph quickly this way...

Given the graph below, find the corresponding function

-2 2 4 6

Read section 3.4 for Friday.

Sec 3.1 and 3 - University of Minnesotahankx003/Fall2012/Lectures/Ch3Sec1an… · Sec 3.1 and 3.3...

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