Analyzing the Graphs of Functions

Objective: To use graphs to make statements about functions.

Finding Domain and Range of a Function

• Use the graph to find:a) The domainb) The rangec) The values of f(-1), f(2)

Finding Domain and Range of a Function

• Use the graph to find:a) The domainb) The rangec) The values of f(-1), f(2)a) Domain = [-1, 5)b) Range = [-3, 3]c) f(-1) = 1; f(2) = -3

Vertical Line Test

• A set of points in a coordinate plane is the graph of y as a function of x if and only if no vertical line intersects the graph at more than one point.

Vertical Line Test

• A set of points in a coordinate plane is the graph of y as a function of x if and only if no vertical line intersects the graph at more than one point.

• We talked about this. A vertical line has the equation x = c. If this line intersects the graph in more than one place, that means for one value of x, there is more than one value for y.

Example 2

• Use the vertical line test to decide whether the graphs represent y as a function of x.

Example 2

• Use the vertical line test to decide whether the graphs represent y as a function of x.

Example 2

• Use the vertical line test to decide whether the graphs represent y as a function of x.

Zeros of a Function

• The zeros of a function f(x) are the x-values for which f(x)=0. This is what we did last chapter when we solved equations for 0. Graphically, we are finding the x-intercepts.

Example 3

• Find the zeros of each function.a) 103)( 2 xxxf

Example 3

• Find the zeros of each function.a)

We need to find the zeros by setting the equation equal to zero and factoring.

103)( 2 xxxf

0)2)(53(

0103 2

xx

xx

2,3

5 xx

Example 3

• Find the zeros of each function.a)

We are now going to find the zeros with our calculator.

103)( 2 xxxf

2,3

5 xx

Example 3

• Find the zeros of each function.b) 210)( xxg

Example 3

• Find the zeros of each function.b) Again, we need to set the equation equal to zero and

solve. A square root is equal to zero when the equation under the radical is equal to zero.

210)( xxg

x

x

x

10

10

0102

2

Example 3

• Find the zeros of each function.b)

Again, we will use our calculator to find the zeros.

210)( xxg

16.310 x

Example 3

• Find the zeros of each function.c)

5

32)(

t

tth

Example 3

• Find the zeros of each function.c)

A fraction is equal to zero when its numerator is equal to zero.

5

32)(

t

tth

2

3

32

032

t

t

t

Example 3

• Find the zeros of each function.c)

Again, let’s use the calculator

5

32)(

t

tth

2

3t

Relative Maximum/Minimum

• A relative Maximum occurs at a peak, or a high point of a graph.

• A relative Minimum occurs at a valley, or a low point of a graph.

Relative Maximum/Minimum

• A relative Maximum occurs at a peak, or a high point of a graph.

• A relative Minimum occurs at a valley, or a low point of a graph.

• The term relative means that this is not the highest or lowest point on the entire graph, just at a certain place.

Relative Maximum/Minimum

• A relative Maximum occurs at a peak, or a high point of a graph.

• A relative Minimum occurs at a valley, or a low point of a graph.

• We will be using our calculators to find these answers.

Increasing/Decreasing

• A function is increasing when it is approaching a relative maximum.

• A function is decreasing as it approaches a relative minimum.

• Again, we will use our calculator to find these answers.

Increasing/Decreasing

• Find where the function is increasing/decreasing.

Increasing/Decreasing

• Find where the function is increasing/decreasing.• This function is increasing everywhere.• Increasing ),(

Increasing/Decreasing

• Find where the function is increasing/decreasing.

Increasing/Decreasing

• Find where the function is increasing/decreasing.• Increasing

• Decreasing

),1(&)1,(

)1,1(

Increasing/Decreasing

• Find where the function is increasing/decreasing.

Increasing/Decreasing

• Find where the function is increasing/decreasing.• Increasing

• Decreasing

• Constant

)0,(

),2(

)2,0(

Example 5

• Use your calculator to find the relative minimum of the function and where the function is increasing or decreasing.

243)( 2 xxxf

Example 5

• Use your calculator to find the relative minimum of the function and where the function is increasing or decreasing.

• So the relative minimum• is at the point (0.67, -3.33).• This function is decreasing and increasing

243)( 2 xxxf

)67.0,(

),67.0(

Example 5

• You try:• Find the relative max and min for the following

function. Then, state where the function is increasing and decreasing. 462)( 23 xxxf

Example 5

• You try:• Find the relative max and min for the following

function. Then, state where the function is increasing and decreasing.

• Max (0, 4)• Min (2, -4)• Increasing• Decreasing

462)( 23 xxxf

),2(&)0,(

)2,0(

Homework

• Pages 210-211• 1-19 odd• 31,33• 49,51,53 (for these, just find max/min and

increasing/decreasing)