1-2 Guided Notes TE - Analyzing Graphs of Functions and Relations

[Pages:6]Name: _________________________________________________ Period: ___________ Date: ________________

Analyzing Graphs of Functions and Relations Guided Notes

The graph of a function is the set of ordered pairs(, ), in the coordinate plane, such that is the domain of .

- the directed distance from the -axis = - the directed distance from the -axis

You can use the graph to estimate function values.

Sample Problem 1: Use a graph of each function to estimate the indicated function values. Then find the values algebraically.

a. = - -

=?

=?

y 5

4

3

2

1

-5 -4 -3 -2 -1 -1

x

1

2

3

4

5

-2

-3

-4

-5

=

=

y 5

4

3

2

1

-5 -4 -3 -2 -1 -1

x

1

2

3

4

5

-2

-3

-4

-5

= - - = - = = - - = - = - =

b. = + +

=?

- =?

- =?

=

- =

- =

y 5

4

3

2

1

-5 -4 -3 -2 -1 -1

x

1

2

3

4

5

-2

-3

-4

-5

y 5

4

3

2

1

-5 -4 -3 -2 -1 -1

x

1

2

3

4

5

-2

-3

-4

-5

= + + = - = - + - + = - + = - = - + - + = - + =

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Name: _________________________________________________ Period: ___________ Date: ________________

Analyzing Graphs of Functions and Relations Guided Notes

Identifying Intercepts from a Functions Graph

A point where the graph intersects or meets the or axis is called an intercept.

An -intercept occurs where = . A -intercept occurs where = .

Sample Problem 2: Use the graph of each function to approximate its ?intercept. Then find the ?intercept algebraically.

a. = -

b. = + +

y 5

4

3

2

1

-5 -4 -3 -2 -1 -1

x

1

2

3

4

5

-2

-3

-4

-5

y 5

4

3

2

1

-5 -4 -3 -2 -1 -1

x

1

2

3

4

5

-2

-3

-4

-5

= -

? =

-intercept occurs where = . = - = - = ? =

= + +

? =

-intercept occurs where = . = + + = ? =

Zeros of a Function

The zeros of function are ?values for which = If the graph of a function of has an -intercept at (, ) then is a zero of the function. To find the zeros of a function, set the function equal to zero and solve for the independent variable.

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Name: _________________________________________________ Period: ___________ Date: ________________

Analyzing Graphs of Functions and Relations Guided Notes

Sample Problem 3: Use the graph of each function to approximate its zeros. Then find the zeros of each function algebraically.

a. = - - =?

= - -

b. = + =?

= +

y 5

4

3

2

1

-5 -4 -3 -2 -1 -1

x

1

2

3

4

5

-2

-3

-4

-5

y 5

4

3

2

1

-5 -4 -3 -2 -1 -1

x

1

2

3

4

5

-2

-3

-4

-5

= - - - -

= - - = - + =

= + = = -

-

= + - .

= + = = - = - - -.

Symmetry of Graphs There are two possible types of symmetry that graphs of functions can have. 1. Line symmetry - graphs can be folded along a line so that the two halves match exactly. 2. Point symmetry - graphs can be rotated 180? with respect to a point and appear unchanged.

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Name: _________________________________________________ Period: ___________ Date: ________________

Analyzing Graphs of Functions and Relations Guided Notes

Tests for Symmetry

Graphical Test

Algebraic Test

The graph of a relation is symmetric with respect to the Replacing with - produces an equivalent -axis if and only if for every point (, ), on the graph, equation. the point (, -), is also on the graph.

The graph of a relation is symmetric with respect to the Replacing with - produces an equivalent -axis if and only if for every point (, )on the graph, equation. the point (-, )is also on the graph.

The graph of a relation is symmetric with respect to the Replacing with - and with - produces an origin if and only if for every point (, ) on the graph, equivalent equation. the point (-, -)is also on the graph.

Sample Problem 4: Use the graph of each equation to test for symmetry with respect to the -axis, -axis, and the origin. Support the answer numerically. Then confirm algebraically.

a.

=

y 5

4

3

2

1

-5 -4 -3 -2 -1 -1

x

1

2

3

4

5

-2

-3

-4

-5

Graphically

The graph appears to be symmetric with respect to the origin because for every point (, ) on the graph, there is a point (-, -).

Support Numerically There is a table of values to support this conjecture.

(, )

-

-

-

-

(-, - )

-

(-, -)

-

(-, -)

(, )

(, )

(, )

Algebraically

- = -

Because - = is equivalent to = , the graph is symmetric

R

with respect to the origin.

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Name: _________________________________________________ Period: ___________ Date: ________________

Analyzing Graphs of Functions and Relations Guided Notes

b. + =

Graphically

y

4

The graph appears to be symmetric with respect to the -axis

3

because for every point (, ) on the graph, there is a point

(, -).

2

1

-4

-3

-2

-1

-1

-2

-3

-4

Support Numerically

x There is a table of values to support this conjecture.

1

2

3

4

-

-

(, )

? ?

?

-, ? (-, ? ) (-, ?)

(, )

Algebraically

+ - = + =

Because + - = is equivalent to + = , the graph is symmetric with respect to the -axis.

Identify Even and Odd Functions If - = , then the function is even, and symmetric to the y-axis. If - = - , then the function is odd, and symmetric to the origin.

Sample Problem 5: Determine whether the following are even, odd, or neither.

a. = +

= + - = - + - = +

- = The function is even.

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Name: _________________________________________________ Period: ___________ Date: ________________

Analyzing Graphs of Functions and Relations Guided Notes

b. = -

= - - = - - - - = - + - = -( - )

- = - The function is odd.

c. = +

= + - = - + (-) - = -

-

- -

The function is neither.

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