title: Describing Graphs

002. Describing Graphs

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Question: `q001. You will frequently need to describe the graphs you have constructed in this course. This exercise is designed to get you used to some of the terminology we use to describe graphs. Please complete this exercise and email your work to the instructor. Note that you should do these graphs on paper without using a calculator. None of the arithmetic involved here should require a calculator, and you should not require the graphing capabilities of your calculator to answer these questions.

Problem 1. We make a table for y = 2x + 7 as follows: We construct two columns, and label the first column 'x' and the second 'y'. Put the numbers -3, -2, -1, -, 1, 2, 3 in the 'x' column. We substitute -3 into the expression and get y = 2(-3) + 7 = 1. We substitute -2 and get y = 2(-2) + 7 = 3. Substituting the remaining numbers we get y values 5, 7, 9, 11 and 13. These numbers go into the second column, each next to the x value from which it was obtained. We then graph these points on a set of x-y coordinate axes. Noting that these points lie on a straight line, we then construct the line through the points.

Now make a table for and graph the function y = 3x - 4.

Identify the intercepts of the graph, i.e., the points where the graph goes through the x and the y axes.

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Your solution:

y = 2x + 7

x | Y

=======

-3 | 1

-2 | 3

-1 | 5

0 | 7

1 | 9

2 |11

3 |13

=======

points form a striaght line

===

y = 3x - 4

x | Y

=======

-3 |-13

-2 |-10

-1 | -7

4/3| 0

0 | -4

1 | -1

2 | 2

3 | 5

=======

y = 3x - 4

y = 3(-3) -4

y = -9 -4

y = -13

y = 3x - 4

y = 3(-2) - 4

y = -6 -4

y = -10

y = 3x - 4

y = 3(-1) - 4

y = -3 -4

y = -7

y = 3x - 4

y = 3(0) - 4

y = 0 - 4

y = -4

y = 3x - 4

y = 3(1) - 4

y = 3 - 4

y = -1

y = 3x - 4

y = 3(2) - 4

y = 6 - 4

y = 2

y = 3x - 4

y = 3(3) - 4

y = 9 - 4

y = 5

y = 3x - 4

0 = 3x - 4

4 = 3x

4/3 = x

This graph is another linear graph. The points are in a straight line. The points that intersect the y axis when x= 0 and y = -4. The points that intersect the x axis when x=4/3 and y = 0

confidence rating #$&*

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Self-critique: OK

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Self-critique rating #$&* 3

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Question: `q002. Does the steepness of the graph in the preceding exercise (of the function y = 3x - 4) change? If so describe how it changes.

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Your solution:

Based on the points that we solved for, it seems that the graph forms a straight line. The slope remained constant at 3/1, which means that the steepness would remain the same.

confidence rating #$&*

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Self-critique: OK

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Self-critique rating #$&* 3

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Question: `q003. What is the slope of the graph of the preceding two exercises (the function is y = 3x - 4;slope is rise / run between two points of the graph)?

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Your solution:

The slope is 3/1

confidence rating #$&*

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Self-critique: OK

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Self-critique rating #$&* 3

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Question: `q004. Make a table of y vs. x for y = x^2. Graph y = x^2 between x = 0 and x = 3.

Would you say that the graph is increasing or decreasing?

Does the steepness of the graph change and if so, how?

Would you say that the graph is increasing at an increasing rate, increasing at a constant rate, increasing at a decreasing rate, decreasing at an decreasing rate, decreasing at a constant rate, or decreasing at a decreasing rate?

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Your solution:

y = x^2

Graph y = x^2 between x = 0 and x = 3.

x | Y

=======

0 | 0

1 | 1

2 | 4

3 | 9

=======

y = x^2

y = 0^2

y = 0

y = x^2

y = 1^2

y = 1

y = x^2

y = 2^2

y = 4

y = x^2

y = 3^2

t = 9

The graph is an exponential graph that is increasing. The slope increases on each given point starting at 0, 1, 1.5, 1.66667 which means that the graph is increasing at an increasing rate with every point set.

confidence rating #$&*

Actually quadratic rather than exponential, but we'll make that distinction soon.

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Self-critique: OK

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Self-critique rating #$&* 3

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Question: `q005. Make a table of y vs. x for y = x^2. Graph y = x^2 between x = -3 and x = 0.

Would you say that the graph is increasing or decreasing?

Does the steepness of the graph change and if so, how?

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Your solution:

y = x^2

Graph y = x^2 between x = -3 and x = 0.

x | Y

=======

-3|9

-2|4

-1|1

0|0

=======

y = x^2

y = -3^2

y = 9

y = x^2

y = -2^2

y = 4

y = x^2

y = -1^2

y = 1

y = x^2

y = 0^2

y = 0

The graph is an exponential graph that is decreasing. The slope decreases on each given point starting at 5, 3, 1 which means that the graph is decreasing at a decreasing rate with every point set.

confidence rating #$&*

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Self-critique: OK

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Self-critique rating #$&* 3

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Question: `q006. Make a table of y vs. x for y = `sqrt(x). [note: `sqrt(x) means 'the square root of x']. Graph y = `sqrt(x) between x = 0 and x = 3.

Would you say that the graph is increasing or decreasing?

Does the steepness of the graph change and if so, how?

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Your solution:

y = `sqrt(x)

Graph y = `sqrt(x) between x = 0 and x = 3

x | Y

=======

0|0

1|1

2|1.41213562

3|1.732050808

=======

y = `sqrt(x)

y = `sqrt(0)

y = 0

y = `sqrt(x)

y = `sqrt(1)

y = 1

y = `sqrt(x)

y = `sqrt(2)

y = 1.41213562

y = `sqrt(x)

y = `sqrt(3)

y = 1.732050808

The graph is increasing at a decreasing rate. The slope decreases on each given point.

confidence rating #$&*

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Self-critique: OK

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Self-critique rating #$&* 3

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Question: `q007. Make a table of y vs. x for y = 5 * 2^(-x). Graph y = 5 * 2^(-x) between x = 0 and x = 3.

Would you say that the graph is increasing or decreasing?

Does the steepness of the graph change and if so, how?

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Your solution:

y = 5 * 2^(-x)

Graph y = 5 * 2^(-x) between x = 0 and x = 3

x | Y

=======

0|5

1|2.5

2|1.25

3|0.625

=======

y = 5 * 2^(-x)

y = 5 * 2^(-0)

y = 5 * 2^(0)

y = 5 * 1

y = 5

y = 5 * 2^(-x)

y = 5 * 2^(-1)

y = 5 * 0.5

y = 2.5

y = 5 * 2^(-x)

y = 5 * 2^(-2)

y = 5 * 0.25

y = 1.25

y = 5 * 2^(-x)

y = 5 * 2^(-3)

y = 5 * 0.125

y = 0.625

The graph is decreasing at a decreasing rate. The slope decreases on each given point.

confidence rating #$&*

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Self-critique: OK

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Self-critique rating #$&* 3

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Question: `q008. Suppose you stand still in front of a driveway. A car starts out next to you and moves away from you, traveling faster and faster.

If y represents the distance from you to the car and t represents the time in seconds since the car started out, would a graph of y vs. t be increasing or decreasing?

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Your solution:

The graph would be increasing at an increasing rate. The further away from me the car got the faster it would be traveling.

confidence rating #$&*

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Self-critique: OK

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Self-critique rating #$&* 3

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Given Solution:

&#Good work. See my notes and let me know if you have questions. &#

#$&*