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College Algebra Tutorial 31: Graphs of Functions, Part I Graphing Functions by Plotting Points
Learning Objectives After completing this tutorial, you should be able to: 1. Graph a function by plotting points. 2. Determine the domain and range of a function from a graph.
Introduction This lesson covers graphing functions by plotting points as well as finding the domain and range of a function after it has been graphed. Graphs are important in giving a visual representation of the correlation between two variables. Even though in this section we are going to look at it generically, using a general x and y variable, you can use two-dimensional graphs for any application where you have two variables. For example, you may have a cost function that is dependent on the quantity of items made. If you needed to show your boss visually the correlation of the quantity with the cost, you could do that on a twodimensional graph. I believe that it is important for you to learn how to do something in general, then when you need to apply it to something specific you have the knowledge to do so. Going from general to specific is a lot easier than specific to general. And that is what we are doing here looking at graphing in general so later you can apply it to something specific, if needed. We will also revisit the concept of domain and range. This time we will be finding them by
looking at a graph. We will be using interval notation to write our answers. In some cases, our domain and/or range will go out to infinity. I think that you are ready to go ahead and graph.
Tutorial
Domain Recall that the domain is the set of all input values to which the rule applies. These are called your independent variables. These are the values that correspond to the first components of the ordered pairs it is associated with. If you need a review on the domain, feel free to go to Tutorial 30: Introduction to Functions. On a graph, the domain corresponds to the horizontal axis. Since that is the case, we need to look to the left and right to see if there are any end points to help us find our domain. If the graph keeps going on and on to the right then the domain is infinity on the right side of the interval. If the graph keeps going on and on to the left then the domain is negative infinity on the left side of the interval.
Range Recall that the range is the set of all output values. These are called your dependent variables. These are the values that correspond to the second components of the ordered pairs it is associated with. If you need a review on the range, feel free to go to Tutorial 30: Introduction to Functions. On a graph, the range corresponds to the vertical axis. Since that is the case, we need to look up and down to see if there are any end points to help us find our range. If the graph keeps going up with no endpoint then the range is infinity on the right side of the interval. If the graph keeps going down then the range goes to negative infinity on the left side of the interval.
Graphing a Function by Plotting Points Step 1: Find at least four ordered pair solutions. Functions can vary on what the graph looks like. So it is good to have a lot of points so that you can get the right shape of the graph, whether it be a straight line, curve, etc..
Step 2: Plot the points found in step 1.
Step 3: Draw the graph. Here are the basic shapes of some of the more common graphs of functions. Keep in mind that these are the basic shapes of these graphs. They can be shifted and stretched depending on the function given. A major goal is to recognize what type of function you are graphing and predict the basic shape from that before you even start. Note that the domain and ranges that go with each one are also given.
Linear Function
Domain: Range:
Constant Function
Domain: Range:
Quadratic Function
Domain: Range:
Cubic Function
Domain: Range:
Square Root Function
Domain: Range:
Absolute Value Function
Domain: Range:
Example 1: Graph the function
using the given values of x. Also use the graph to determine the domain and range of the function.
x = -3, -2, -1, 0, 1, 2, 3
Step 1: Find at least four ordered pair solutions. I’m going to use a chart to organize my information. A chart keeps track of the x values that you are using and the corresponding y value found when you used a particular x value. We will use the seven values of x that were given to find corresponding functional values to give us seven ordered pair solutions. Keep in mind that the functional value that we find correlates to the second or y value of our ordered pair.
x
(x , y )
-3
(-3, 4)
-2
(-2, -1)
-1
(-1, - 4)
0
(0, -5)
1
(1, - 4)
2
(2, -1)
3
(3, 4)
Step 2: Plot the points found in step 1.
(-3,4)
(-2,-1) (-1,-4) (0,-5) (1,-4) (2,-1) (3,4)
Step 3: Draw the graph. This function is a quadratic function, so its graph will be a parabola.
(-3,4)
(-2,-1) (-1,-4) (0,-5) (1,-4) (2,-1) (3,4)
Domain We need to find the set of all input values. In terms of ordered pairs, that correlates with the first component of each one. In terms of this two dimensional graph, that corresponds with the x values (horizontal axis). Since that is the case, we need to look to the left and right and see if there are any end points. In this case, note how the curve has arrows at both ends, that means it would go on and on forever to the right and to the left. This means that the domain is
.
Range We need to find the set of all output values. In terms of ordered pairs, that correlates with the second component of each one. In terms of this two dimensional graph, that corresponds with the y values (vertical axis). Since that is the case, we need to look up and down and see if there are any end points. In this case, note how the curve has a low endpoint of y = -5 and it has arrows at both ends going up, that means it would go up on and on forever. This means that the range is
.
Example 2: Graph the function
using the given values of x. Also use the graph to determine the domain and range of the function.
x = 0, 1, 4, 9
Step 1: Find at least four ordered pair solutions. I’m going to use a chart to organize my information. A chart keeps track of the x values that you are using and the corresponding y value found when you used a particular x value. We will use the four values of x that were given to find corresponding functional values to give us four ordered pair solutions. Keep in mind that the functional value that we find correlates to the second or y value of our ordered pair.
x
(x , y )
0
(0, 3)
1
(1, 4)
4
(4, 5)
9
(9, 6)
Step 2: Plot the points found in step 1.
Step 3: Draw the graph. This function is a square root function, so its graph will be a curve.
Domain We need to find the set of all input values. In terms of ordered pairs, that correlates with the first component of each one. In terms of this two dimensional graph, that corresponds with the x values (horizontal axis). Since that is the case, we need to look to the left and right and see if there are any end points. In this case, note how there is a left endpoint at x = 0. Also note that the curve has an arrow going to the right, that means it would go on and on forever to the right. This means that the domain is
.
Range We need to find the set of all output values. In terms of ordered pairs, that correlates with the second component of each one. In terms of this two dimensional graph, that corresponds with the y values (vertical axis). Since that is the case, we need to look up and down and see if there are any end points. In this case, note how the curve has a low endpoint of y = 3 and it has an arrow going up, that means it would go on and on forever up. This means that the range is
.
Example 3: Graph the function
using the given values of x. Also use the graph to determine the domain and range of the function.
x = 0, 1, 2, 3, 4, 5, 6
Step 1: Find at least four ordered pair solutions.
I’m going to use a chart to organize my information. A chart keeps track of the x values that you are using and the corresponding y value found when you used a particular x value. We will use the seven values of x that were given to find corresponding functional values to give us seven ordered pair solutions. Keep in mind that the functional value that we find correlates to the second or y value of our ordered pair.
x
(x, y)
0
(0, 4)
1
(1, 3)
2
(2, 2)
3
(3, 1)
4
(4, 0)
5
(5, 1)
6
(6, 2)
Step 2: Plot the points found in step 1.
Step 3: Draw the graph.
This function is an absolute value function, so its graph will be a v shape.
Domain We need to find the set of all input values. In terms of ordered pairs, that correlates with the first component of each one. In terms of this two dimensional graph, that corresponds with the x values (horizontal axis). Since that is the case, we need to look to the left and right and see if there are any end points. In this case, note how there are arrows at both ends, that means it would go on and on forever to the right and to the left. This means that the domain is
.
Range We need to find the set of all output values. In terms of ordered pairs, that correlates with the second component of each one. In terms of this two dimensional graph, that corresponds with the y values (vertical axis). Since that is the case, we need to look up and down and see if there are any end points. In this case, note how the graph has a low endpoint of
y = 0 and it has an arrows that go up, that means it would go up on and on forever. This means that the range is
.
Example 4: Graph the function
using the given values of x. Also use the graph to determine the domain and range of the function.
x = -3, -2, -1, 0, 1, 2, 3
Step 1: Find at least four ordered pair solutions. I’m going to use a chart to organize my information. A chart keeps track of the x values that you are using and the corresponding y value found when you used a particular x value. We will use the seven values of x that were given to find corresponding functional values to give us seven ordered pair solutions. Keep in mind that the functional value that we find correlates to the second or y value of our ordered pair.
x
(x, y)
-3
(-3, -2)
-2
(-2, -2)
-1
(-1, -2)
0
(0, -2)
1
(1, -2)
2
(2, -2)
3
(3, -2)
Step 2: Plot the points found in step 1.
Step 3: Draw the graph. This function is a constant function, so its graph will be a horizontal line.
Domain We need to find the set of all input values. In terms of ordered pairs, that correlates with the first component of each one. In terms of this two dimensional graph, that corresponds with the x values (horizontal
axis). Since that is the case, we need to look to the left and right and see if there are any end points. In this case, note how the curve has arrows at both ends, that means it would go on and on forever to the right and to the left. This means that the domain is
.
Range We need to find the set of all output values. In terms of ordered pairs, that correlates with the second component of each one. In terms of this two dimensional graph, that corresponds with the y values (vertical axis). Since that is the case, we need to look up and down and see if there are any end points. In this case, note how the line does not go up or down and that all the values of y are -2. This means that the range is {y | y = -2}.
Example 5: Graph the function
using the given values of x. Also use the graph to determine the domain and range of the function.
x = -2, -1, 0, 1, 2
Step 1: Find at least four ordered pair solutions. I’m going to use a chart to organize my information. A chart keeps track of the x values that you are using and the corresponding y value found when you used a particular x value. We will use the five values of x that were given to find corresponding
functional values to give us five ordered pair solutions. Keep in mind that the functional value that we find correlates to the second or y value of our ordered pair.
x
(x, y)
-2
(-2, -11)
-1
(-1, - 4)
0
(0, -3)
1
(1, -2)
2
(2, 5)
Step 2: Plot the points found in step 1.
Step 3: Draw the graph. This function is a cubic function, so its graph will be a curve.
Domain We need to find the set of all input values. In terms of ordered pairs, that correlates with the first component of each one. In terms of this two dimensional graph, that corresponds with the x values (horizontal axis). Since that is the case, we need to look to the left and right and see if there are any end points. In this case, note how the curve has arrows at both ends, that means it would go on and on forever to the right and to the left. This means that the domain is
.
Range We need to find the set of all output values. In terms of ordered pairs, that correlates with the second component of each one. In terms of this two dimensional graph, that corresponds with the y values (vertical axis). Since that is the case, we need to look up and down and see if there are any end points. In this case, note how the curve has arrows at both ends, that means it would go on and on forever up and down.
This means that the range is
.
Practice Problems These are practice problems to help bring you to the next level. It will allow you to check and see if you have an understanding of these types of problems. Math works just like anything else, if you want to get good at it, then you need to practice it. Even the best athletes and musicians had help along the way and lots of practice, practice, practice, to get good at their sport or instrument. In fact there is no such thing as too much practice. To get the most out of these, you should work the problem out on your own and then check your answer by clicking on the link for the answer/discussion for that problem. At the link you will find the answer as well as any steps that went into finding that answer.
Practice Problems 1a - 1e: Graph the given function using the given values of x. Also use the graph to determine the domain and range of the function.
1a.
; x = -3, -2, -1, 0, 1, 2, 3
(answer/discussion to 1a)
1c.
1b.
; x = -3, -2, -1, 0, 1, 2, 3 1d.
(answer/discussion to 1c)
; x = -3, -2, 1, 6
(answer/discussion to 1b)
; x = -3, -2, -1, 0, 1, 2, 3
(answer/discussion to 1d)
1e.
; x = -2, -1, 0, 1, 2
(answer/discussion to 1e)
Need Extra Help on These Topics?
The following are webpages that can assist you in the topics that were covered on this page: http://www.wtamu.edu/academic/anns/mps/math/mathlab/int_algebra / int_alg_tut12_graph.htm This webpage helps you with graphing equations by plotting points.
Go to Get Help Outside the Classroom found in Tutorial 1: How to Succeed in a Math Class for some more suggestions.
(Back to the College Algebra Homepage) All contents copyright (C) 2002, WTAMU and Kim Peppard. All rights reserved. Last revised on September 6, 2002 by Kim Peppard.
College Algebra Tutorial 31: Graphs of Functions, Part I Graphing Functions by Plotting Points
Learning Objectives After completing this tutorial, you should be able to: 1. Graph a function by plotting points. 2. Determine the domain and range of a function from a graph.
Introduction This lesson covers graphing functions by plotting points as well as finding the domain and range of a function after it has been graphed. Graphs are important in giving a visual representation of the correlation between two variables. Even though in this section we are going to look at it generically, using a general x and y variable, you can use two-dimensional graphs for any application where you have two variables. For example, you may have a cost function that is dependent on the quantity of items made. If you needed to show your boss visually the correlation of the quantity with the cost, you could do that on a twodimensional graph. I believe that it is important for you to learn how to do something in general, then when you need to apply it to something specific you have the knowledge to do so. Going from general to specific is a lot easier than specific to general. And that is what we are doing here looking at graphing in general so later you can apply it to something specific, if needed. We will also revisit the concept of domain and range. This time we will be finding them by
looking at a graph. We will be using interval notation to write our answers. In some cases, our domain and/or range will go out to infinity. I think that you are ready to go ahead and graph.
Tutorial
Domain Recall that the domain is the set of all input values to which the rule applies. These are called your independent variables. These are the values that correspond to the first components of the ordered pairs it is associated with. If you need a review on the domain, feel free to go to Tutorial 30: Introduction to Functions. On a graph, the domain corresponds to the horizontal axis. Since that is the case, we need to look to the left and right to see if there are any end points to help us find our domain. If the graph keeps going on and on to the right then the domain is infinity on the right side of the interval. If the graph keeps going on and on to the left then the domain is negative infinity on the left side of the interval.
Range Recall that the range is the set of all output values. These are called your dependent variables. These are the values that correspond to the second components of the ordered pairs it is associated with. If you need a review on the range, feel free to go to Tutorial 30: Introduction to Functions. On a graph, the range corresponds to the vertical axis. Since that is the case, we need to look up and down to see if there are any end points to help us find our range. If the graph keeps going up with no endpoint then the range is infinity on the right side of the interval. If the graph keeps going down then the range goes to negative infinity on the left side of the interval.
Graphing a Function by Plotting Points Step 1: Find at least four ordered pair solutions. Functions can vary on what the graph looks like. So it is good to have a lot of points so that you can get the right shape of the graph, whether it be a straight line, curve, etc..
Step 2: Plot the points found in step 1.
Step 3: Draw the graph. Here are the basic shapes of some of the more common graphs of functions. Keep in mind that these are the basic shapes of these graphs. They can be shifted and stretched depending on the function given. A major goal is to recognize what type of function you are graphing and predict the basic shape from that before you even start. Note that the domain and ranges that go with each one are also given.
Linear Function
Domain: Range:
Constant Function
Domain: Range:
Quadratic Function
Domain: Range:
Cubic Function
Domain: Range:
Square Root Function
Domain: Range:
Absolute Value Function
Domain: Range:
Example 1: Graph the function
using the given values of x. Also use the graph to determine the domain and range of the function.
x = -3, -2, -1, 0, 1, 2, 3
Step 1: Find at least four ordered pair solutions. I’m going to use a chart to organize my information. A chart keeps track of the x values that you are using and the corresponding y value found when you used a particular x value. We will use the seven values of x that were given to find corresponding functional values to give us seven ordered pair solutions. Keep in mind that the functional value that we find correlates to the second or y value of our ordered pair.
x
(x , y )
-3
(-3, 4)
-2
(-2, -1)
-1
(-1, - 4)
0
(0, -5)
1
(1, - 4)
2
(2, -1)
3
(3, 4)
Step 2: Plot the points found in step 1.
(-3,4)
(-2,-1) (-1,-4) (0,-5) (1,-4) (2,-1) (3,4)
Step 3: Draw the graph. This function is a quadratic function, so its graph will be a parabola.
(-3,4)
(-2,-1) (-1,-4) (0,-5) (1,-4) (2,-1) (3,4)
Domain We need to find the set of all input values. In terms of ordered pairs, that correlates with the first component of each one. In terms of this two dimensional graph, that corresponds with the x values (horizontal axis). Since that is the case, we need to look to the left and right and see if there are any end points. In this case, note how the curve has arrows at both ends, that means it would go on and on forever to the right and to the left. This means that the domain is
.
Range We need to find the set of all output values. In terms of ordered pairs, that correlates with the second component of each one. In terms of this two dimensional graph, that corresponds with the y values (vertical axis). Since that is the case, we need to look up and down and see if there are any end points. In this case, note how the curve has a low endpoint of y = -5 and it has arrows at both ends going up, that means it would go up on and on forever. This means that the range is
.
Example 2: Graph the function
using the given values of x. Also use the graph to determine the domain and range of the function.
x = 0, 1, 4, 9
Step 1: Find at least four ordered pair solutions. I’m going to use a chart to organize my information. A chart keeps track of the x values that you are using and the corresponding y value found when you used a particular x value. We will use the four values of x that were given to find corresponding functional values to give us four ordered pair solutions. Keep in mind that the functional value that we find correlates to the second or y value of our ordered pair.
x
(x , y )
0
(0, 3)
1
(1, 4)
4
(4, 5)
9
(9, 6)
Step 2: Plot the points found in step 1.
Step 3: Draw the graph. This function is a square root function, so its graph will be a curve.
Domain We need to find the set of all input values. In terms of ordered pairs, that correlates with the first component of each one. In terms of this two dimensional graph, that corresponds with the x values (horizontal axis). Since that is the case, we need to look to the left and right and see if there are any end points. In this case, note how there is a left endpoint at x = 0. Also note that the curve has an arrow going to the right, that means it would go on and on forever to the right. This means that the domain is
.
Range We need to find the set of all output values. In terms of ordered pairs, that correlates with the second component of each one. In terms of this two dimensional graph, that corresponds with the y values (vertical axis). Since that is the case, we need to look up and down and see if there are any end points. In this case, note how the curve has a low endpoint of y = 3 and it has an arrow going up, that means it would go on and on forever up. This means that the range is
.
Example 3: Graph the function
using the given values of x. Also use the graph to determine the domain and range of the function.
x = 0, 1, 2, 3, 4, 5, 6
Step 1: Find at least four ordered pair solutions.
I’m going to use a chart to organize my information. A chart keeps track of the x values that you are using and the corresponding y value found when you used a particular x value. We will use the seven values of x that were given to find corresponding functional values to give us seven ordered pair solutions. Keep in mind that the functional value that we find correlates to the second or y value of our ordered pair.
x
(x, y)
0
(0, 4)
1
(1, 3)
2
(2, 2)
3
(3, 1)
4
(4, 0)
5
(5, 1)
6
(6, 2)
Step 2: Plot the points found in step 1.
Step 3: Draw the graph.
This function is an absolute value function, so its graph will be a v shape.
Domain We need to find the set of all input values. In terms of ordered pairs, that correlates with the first component of each one. In terms of this two dimensional graph, that corresponds with the x values (horizontal axis). Since that is the case, we need to look to the left and right and see if there are any end points. In this case, note how there are arrows at both ends, that means it would go on and on forever to the right and to the left. This means that the domain is
.
Range We need to find the set of all output values. In terms of ordered pairs, that correlates with the second component of each one. In terms of this two dimensional graph, that corresponds with the y values (vertical axis). Since that is the case, we need to look up and down and see if there are any end points. In this case, note how the graph has a low endpoint of
y = 0 and it has an arrows that go up, that means it would go up on and on forever. This means that the range is
.
Example 4: Graph the function
using the given values of x. Also use the graph to determine the domain and range of the function.
x = -3, -2, -1, 0, 1, 2, 3
Step 1: Find at least four ordered pair solutions. I’m going to use a chart to organize my information. A chart keeps track of the x values that you are using and the corresponding y value found when you used a particular x value. We will use the seven values of x that were given to find corresponding functional values to give us seven ordered pair solutions. Keep in mind that the functional value that we find correlates to the second or y value of our ordered pair.
x
(x, y)
-3
(-3, -2)
-2
(-2, -2)
-1
(-1, -2)
0
(0, -2)
1
(1, -2)
2
(2, -2)
3
(3, -2)
Step 2: Plot the points found in step 1.
Step 3: Draw the graph. This function is a constant function, so its graph will be a horizontal line.
Domain We need to find the set of all input values. In terms of ordered pairs, that correlates with the first component of each one. In terms of this two dimensional graph, that corresponds with the x values (horizontal
axis). Since that is the case, we need to look to the left and right and see if there are any end points. In this case, note how the curve has arrows at both ends, that means it would go on and on forever to the right and to the left. This means that the domain is
.
Range We need to find the set of all output values. In terms of ordered pairs, that correlates with the second component of each one. In terms of this two dimensional graph, that corresponds with the y values (vertical axis). Since that is the case, we need to look up and down and see if there are any end points. In this case, note how the line does not go up or down and that all the values of y are -2. This means that the range is {y | y = -2}.
Example 5: Graph the function
using the given values of x. Also use the graph to determine the domain and range of the function.
x = -2, -1, 0, 1, 2
Step 1: Find at least four ordered pair solutions. I’m going to use a chart to organize my information. A chart keeps track of the x values that you are using and the corresponding y value found when you used a particular x value. We will use the five values of x that were given to find corresponding
functional values to give us five ordered pair solutions. Keep in mind that the functional value that we find correlates to the second or y value of our ordered pair.
x
(x, y)
-2
(-2, -11)
-1
(-1, - 4)
0
(0, -3)
1
(1, -2)
2
(2, 5)
Step 2: Plot the points found in step 1.
Step 3: Draw the graph. This function is a cubic function, so its graph will be a curve.
Domain We need to find the set of all input values. In terms of ordered pairs, that correlates with the first component of each one. In terms of this two dimensional graph, that corresponds with the x values (horizontal axis). Since that is the case, we need to look to the left and right and see if there are any end points. In this case, note how the curve has arrows at both ends, that means it would go on and on forever to the right and to the left. This means that the domain is
.
Range We need to find the set of all output values. In terms of ordered pairs, that correlates with the second component of each one. In terms of this two dimensional graph, that corresponds with the y values (vertical axis). Since that is the case, we need to look up and down and see if there are any end points. In this case, note how the curve has arrows at both ends, that means it would go on and on forever up and down.
This means that the range is
.
Practice Problems These are practice problems to help bring you to the next level. It will allow you to check and see if you have an understanding of these types of problems. Math works just like anything else, if you want to get good at it, then you need to practice it. Even the best athletes and musicians had help along the way and lots of practice, practice, practice, to get good at their sport or instrument. In fact there is no such thing as too much practice. To get the most out of these, you should work the problem out on your own and then check your answer by clicking on the link for the answer/discussion for that problem. At the link you will find the answer as well as any steps that went into finding that answer.
Practice Problems 1a - 1e: Graph the given function using the given values of x. Also use the graph to determine the domain and range of the function.
1a.
; x = -3, -2, -1, 0, 1, 2, 3
(answer/discussion to 1a)
1c.
1b.
; x = -3, -2, -1, 0, 1, 2, 3 1d.
(answer/discussion to 1c)
; x = -3, -2, 1, 6
(answer/discussion to 1b)
; x = -3, -2, -1, 0, 1, 2, 3
(answer/discussion to 1d)
1e.
; x = -2, -1, 0, 1, 2
(answer/discussion to 1e)
Need Extra Help on These Topics?
The following are webpages that can assist you in the topics that were covered on this page: http://www.wtamu.edu/academic/anns/mps/math/mathlab/int_algebra / int_alg_tut12_graph.htm This webpage helps you with graphing equations by plotting points.
Go to Get Help Outside the Classroom found in Tutorial 1: How to Succeed in a Math Class for some more suggestions.
(Back to the College Algebra Homepage) All contents copyright (C) 2002, WTAMU and Kim Peppard. All rights reserved. Last revised on September 6, 2002 by Kim Peppard.
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