Analyzing the Graphs of Functions Objective: To use graphs to make statements about functions.
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Transcript of Analyzing the Graphs of Functions Objective: To use graphs to make statements about functions.
![Page 1: Analyzing the Graphs of Functions Objective: To use graphs to make statements about functions.](https://reader036.fdocuments.us/reader036/viewer/2022062421/56649dde5503460f94ad7882/html5/thumbnails/1.jpg)
Analyzing the Graphs of Functions
Objective: To use graphs to make statements about functions.
![Page 2: Analyzing the Graphs of Functions Objective: To use graphs to make statements about functions.](https://reader036.fdocuments.us/reader036/viewer/2022062421/56649dde5503460f94ad7882/html5/thumbnails/2.jpg)
Finding Domain and Range of a Function
• Use the graph to find:a) The domainb) The rangec) The values of f(-1), f(2)
![Page 3: Analyzing the Graphs of Functions Objective: To use graphs to make statements about functions.](https://reader036.fdocuments.us/reader036/viewer/2022062421/56649dde5503460f94ad7882/html5/thumbnails/3.jpg)
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
![Page 4: Analyzing the Graphs of Functions Objective: To use graphs to make statements about functions.](https://reader036.fdocuments.us/reader036/viewer/2022062421/56649dde5503460f94ad7882/html5/thumbnails/4.jpg)
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.
![Page 5: Analyzing the Graphs of Functions Objective: To use graphs to make statements about functions.](https://reader036.fdocuments.us/reader036/viewer/2022062421/56649dde5503460f94ad7882/html5/thumbnails/5.jpg)
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.
![Page 6: Analyzing the Graphs of Functions Objective: To use graphs to make statements about functions.](https://reader036.fdocuments.us/reader036/viewer/2022062421/56649dde5503460f94ad7882/html5/thumbnails/6.jpg)
Example 2
• Use the vertical line test to decide whether the graphs represent y as a function of x.
![Page 7: Analyzing the Graphs of Functions Objective: To use graphs to make statements about functions.](https://reader036.fdocuments.us/reader036/viewer/2022062421/56649dde5503460f94ad7882/html5/thumbnails/7.jpg)
Example 2
• Use the vertical line test to decide whether the graphs represent y as a function of x.
![Page 8: Analyzing the Graphs of Functions Objective: To use graphs to make statements about functions.](https://reader036.fdocuments.us/reader036/viewer/2022062421/56649dde5503460f94ad7882/html5/thumbnails/8.jpg)
Example 2
• Use the vertical line test to decide whether the graphs represent y as a function of x.
![Page 9: Analyzing the Graphs of Functions Objective: To use graphs to make statements about functions.](https://reader036.fdocuments.us/reader036/viewer/2022062421/56649dde5503460f94ad7882/html5/thumbnails/9.jpg)
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.
![Page 10: Analyzing the Graphs of Functions Objective: To use graphs to make statements about functions.](https://reader036.fdocuments.us/reader036/viewer/2022062421/56649dde5503460f94ad7882/html5/thumbnails/10.jpg)
Example 3
• Find the zeros of each function.a) 103)( 2 xxxf
![Page 11: Analyzing the Graphs of Functions Objective: To use graphs to make statements about functions.](https://reader036.fdocuments.us/reader036/viewer/2022062421/56649dde5503460f94ad7882/html5/thumbnails/11.jpg)
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
![Page 12: Analyzing the Graphs of Functions Objective: To use graphs to make statements about functions.](https://reader036.fdocuments.us/reader036/viewer/2022062421/56649dde5503460f94ad7882/html5/thumbnails/12.jpg)
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
![Page 13: Analyzing the Graphs of Functions Objective: To use graphs to make statements about functions.](https://reader036.fdocuments.us/reader036/viewer/2022062421/56649dde5503460f94ad7882/html5/thumbnails/13.jpg)
Example 3
• Find the zeros of each function.b) 210)( xxg
![Page 14: Analyzing the Graphs of Functions Objective: To use graphs to make statements about functions.](https://reader036.fdocuments.us/reader036/viewer/2022062421/56649dde5503460f94ad7882/html5/thumbnails/14.jpg)
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
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Example 3
• Find the zeros of each function.b)
Again, we will use our calculator to find the zeros.
210)( xxg
16.310 x
![Page 16: Analyzing the Graphs of Functions Objective: To use graphs to make statements about functions.](https://reader036.fdocuments.us/reader036/viewer/2022062421/56649dde5503460f94ad7882/html5/thumbnails/16.jpg)
Example 3
• Find the zeros of each function.c)
5
32)(
t
tth
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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
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Example 3
• Find the zeros of each function.c)
Again, let’s use the calculator
5
32)(
t
tth
2
3t
![Page 19: Analyzing the Graphs of Functions Objective: To use graphs to make statements about functions.](https://reader036.fdocuments.us/reader036/viewer/2022062421/56649dde5503460f94ad7882/html5/thumbnails/19.jpg)
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.
![Page 20: Analyzing the Graphs of Functions Objective: To use graphs to make statements about functions.](https://reader036.fdocuments.us/reader036/viewer/2022062421/56649dde5503460f94ad7882/html5/thumbnails/20.jpg)
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.
![Page 21: Analyzing the Graphs of Functions Objective: To use graphs to make statements about functions.](https://reader036.fdocuments.us/reader036/viewer/2022062421/56649dde5503460f94ad7882/html5/thumbnails/21.jpg)
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.
![Page 22: Analyzing the Graphs of Functions Objective: To use graphs to make statements about functions.](https://reader036.fdocuments.us/reader036/viewer/2022062421/56649dde5503460f94ad7882/html5/thumbnails/22.jpg)
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.
![Page 23: Analyzing the Graphs of Functions Objective: To use graphs to make statements about functions.](https://reader036.fdocuments.us/reader036/viewer/2022062421/56649dde5503460f94ad7882/html5/thumbnails/23.jpg)
Increasing/Decreasing
• Find where the function is increasing/decreasing.
![Page 24: Analyzing the Graphs of Functions Objective: To use graphs to make statements about functions.](https://reader036.fdocuments.us/reader036/viewer/2022062421/56649dde5503460f94ad7882/html5/thumbnails/24.jpg)
Increasing/Decreasing
• Find where the function is increasing/decreasing.• This function is increasing everywhere.• Increasing ),(
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Increasing/Decreasing
• Find where the function is increasing/decreasing.
![Page 26: Analyzing the Graphs of Functions Objective: To use graphs to make statements about functions.](https://reader036.fdocuments.us/reader036/viewer/2022062421/56649dde5503460f94ad7882/html5/thumbnails/26.jpg)
Increasing/Decreasing
• Find where the function is increasing/decreasing.• Increasing
• Decreasing
),1(&)1,(
)1,1(
![Page 27: Analyzing the Graphs of Functions Objective: To use graphs to make statements about functions.](https://reader036.fdocuments.us/reader036/viewer/2022062421/56649dde5503460f94ad7882/html5/thumbnails/27.jpg)
Increasing/Decreasing
• Find where the function is increasing/decreasing.
![Page 28: Analyzing the Graphs of Functions Objective: To use graphs to make statements about functions.](https://reader036.fdocuments.us/reader036/viewer/2022062421/56649dde5503460f94ad7882/html5/thumbnails/28.jpg)
Increasing/Decreasing
• Find where the function is increasing/decreasing.• Increasing
• Decreasing
• Constant
)0,(
),2(
)2,0(
![Page 29: Analyzing the Graphs of Functions Objective: To use graphs to make statements about functions.](https://reader036.fdocuments.us/reader036/viewer/2022062421/56649dde5503460f94ad7882/html5/thumbnails/29.jpg)
Example 5
• Use your calculator to find the relative minimum of the function and where the function is increasing or decreasing.
243)( 2 xxxf
![Page 30: Analyzing the Graphs of Functions Objective: To use graphs to make statements about functions.](https://reader036.fdocuments.us/reader036/viewer/2022062421/56649dde5503460f94ad7882/html5/thumbnails/30.jpg)
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(
![Page 31: Analyzing the Graphs of Functions Objective: To use graphs to make statements about functions.](https://reader036.fdocuments.us/reader036/viewer/2022062421/56649dde5503460f94ad7882/html5/thumbnails/31.jpg)
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
![Page 32: Analyzing the Graphs of Functions Objective: To use graphs to make statements about functions.](https://reader036.fdocuments.us/reader036/viewer/2022062421/56649dde5503460f94ad7882/html5/thumbnails/32.jpg)
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(
![Page 33: Analyzing the Graphs of Functions Objective: To use graphs to make statements about functions.](https://reader036.fdocuments.us/reader036/viewer/2022062421/56649dde5503460f94ad7882/html5/thumbnails/33.jpg)
Homework
• Pages 210-211• 1-19 odd• 31,33• 49,51,53 (for these, just find max/min and
increasing/decreasing)