1-2: Analyzing Graphs of Functions and Relations

Realtors in a metropolitan area student the average home price per square foot as a function of total square footage. Their evaluation yielded the following piecewise-defined function. Find the average price per square foot for a home with the given square footage.

p

a

a

1000 75 40

a 2600 110

100

a

4000 25

98

If 1000

2600

If 2600

4000

If 4000

a). 1400 square feet

b). 3200 square feet

p 1400 1400 1000 75

40 $85

p 3200 3200 2600 110

100 $104

1-2: Analyzing Graphs of Functions and Relations

Precalculus Mr. Gallo

1

Analyzing Function Graphs

You should be able to: 1. Estimate a value from the graph. 2. Find the domain and range from the graph. 3. Find the y-intercepts. 4. Find the zeroes from the graph.

The function

5 50 approximates the profit at a toy company,

where is the marketing costs and

is profit. Both costs and profits are

measured in tens of thousands of dollars.

a. Use the graph to estimate the profit when marketing costs are $30,000. Confirm your estimate algebraically.

150

f x = -5x2+50x

Profit ($10,000)

f 30, 000 $1, 050, 000

100

b. What is the domain and range of the

function?

50

D: 0,100, 000 R: 0,125, 000

c. Use the graph to estimate the y-intercept of the function. Confirm your estimate algebraically. y0

5

10

Marketing ($10,000)

d. Use the graph to estimate the zeroes of the function. Confirm your estimate algebraically. x 0 and x 100, 000

2

Types of Symmetries

Line Symmetry Can be folded along a line so the two halves match exactly

Point Symmetry Can be rotated 180? with respect to a point and appear unchanged.

Three types of symmetries: 1. With respect to the axis 2. With respect to the axis 3. With respect to the origin

Tests for Symmetry (on p.16 in book)

Graphical Test

Symmetric with respect to the x-axis iff. for every point

, on the graph, the point , is also on the graph Symmetric with respect to the y-axis iff. for every point

, on the graph, the point , is also on the graph. Symmetric with respect to the origin iff. for every point

, on the graph, the point , is also on the graph.

Model

(x,y ) y

x (x,-y )

(-x,y ) y

x (x,y )

(x,y )

(-x,-y ) -2

Algebraic Test

Replacing with produces an equivalent equation.

Replacing with produces an equivalent equation.

Replacing with and with produces an equivalent equation.

3

Show that

is symmetric to the -axis-graphically and algebraically.

If symmetric to the -axis, then

Graphically:

4

(-2,4)

(2,4)

2

will produced the same value as .

Algebraically:

y x2

f x x2 1x2

f x x2

12 x2

1x2

x2

The answers are the same so

is

symmetric with respect to the -axis.

Show that

is symmetric to the -axis-graphically and algebraically.

If symmetric to the -axis, then

Graphically:

4

(3,3)

2

5 -2

(3,-3)

-4

will produced the same value as .

Algebraically:

x y

f y y

x y

f y y

The answers are the same so is symmetric with respect to the axis.

4

Show that

is symmetric to the origin-graphically and algebraically.

If symmetric to the origin, then

Graphically:

2

(1,1)

(-1,-1)

-2

will produced

.

Algebraically:

y x y x3

3

f x x3

f x x3

The answers are the opposites so is symmetric with respect to the origin.

Show that 2

is symmetric to the origin-algebraically.

If symmetric to the origin, then

will produced

.

Algebraically:

f x 2x5 x3 x

f x 2x5 x3 x

2x5 x3 x

2x5 x3 x

factor out a -1:

2x5 x3 x

The answers are the opposites so 2 respect to the origin.

is symmetric with

5

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