assignment 5

course Mth 163

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assignment #005

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00:00:27

`q001. Note that this assignment has 8 questions

Evaluate the function y = x^2 for x values -3, -2, -1, 0, 1, 2, and 3. What are your y values?

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y=x^2

x y

-3 9

-2 4

-1 1

0 0

1 1

2 4

3 9

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00:00:37

You should have obtained y values 9, 4, 1, 0, 1, 4, 9, in that order.

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00:11:07

`q002. Evaluate the function y = 2^x for x values -3, -2, -1, 0, 1, 2, and 3. What are your y values?

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y values are .125, .25, .5, 1, 2, 4, 8

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00:11:55

By velocity exponents, b^-x = 1 / b^x. So for example 2^-2 = 1 / 2^2 = 1/4. Your y values will be 1/8, 1/4, 1/2, 1, 2, 4 and 8. Note that we have used the fact that for any b, b^0 = 1. It is a common error to say that 2^0 is 0. Note that this error would interfere with the pattern or progression of the y values.

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00:16:09

`q003. Evaluate the function y = x^-2 for x values -3, -2, -1, 0, 1, 2, and 3. What are your y values?

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y values are 1/9, 1/4, 1, error, 1, 1/4, 1/9

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00:16:32

By the laws of exponents, x^-p = 1 / x^p. So x^-2 = 1 / x^2, and your x values should be 1/9, 1/4, and 1. Since 1 / 0^2 = 1 / 0 and division by zero is not defined, the x = 0 value is undefined. The last three values will be 1, 1/4, and 1/9.

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00:17:41

`q004. Evaluate the function y = x^3 for x values -3, -2, -1, 0, 1, 2, and 3. What are your y values?

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y values are -27, -8, -1, 0, 1, 8, 27

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00:18:27

The y values should be -27, -8, -1, 0, 1, 4, 9.

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The responses for 2^3 and 3^3 should be 8 and 27.

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00:26:21

`q005. Sketch graphs for y = x^2, y = 2^x, y = x^-2 and y = x^3, using the values you obtained in the preceding four problems. Describe the graph of each function.

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y=x^2 is a parabola with vertex at the origin

y=2^x is a upward increasing graph with a horizontal asymptote at x=0

y=x^-2 is a graph with an upward increasing slop with horizontal and vertical asymptotes at y=0 and x=0

y=x^3 is an s shaped graph switching directions at 0

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00:27:59

The graph of y = x^2 is a parabola with its vertex at the origin. It is worth noting that the graph is symmetric with respect to the y-axis. That is, the graph to the left of the y-axis is a mirror image of the graph to the right of the y-axis.

The graph of y = 2^x begins at x = -3 with value 1/8, which is relatively close to zero. The graph therefore starts to the left, close to the x-axis. With each succeeding unit of x, with x moving to the right, the y value doubles. This causes the graph to rise more and more quickly as we move from left to right. The graph intercepts the y-axis at y = 1.

The graph of y = x^-2 rises more and more rapidly as we approach the y-axis from the left. It might not be clear from the values obtained here that this progression continues, with the y values increasing beyond bound, but this is the case. This behavior is mirrored on the other side of the y-axis, so that the graph rises as we approach the y-axis from either side. In fact the graph rises without bound as we approach the y-axis from either side. The y-axis is therefore a vertical asymptote for this graph.

The graph of y = x ^ 3 has negative y values whenever x is negative and positive y values whenever x is positive. As we approach x = 0 from the left, through negative x values, the y values increase toward zero, but the rate of increase slows so that the graph actually levels off for an instant at the point (0,0) before beginning to increase again. To the right of x = 0 the graph increases faster and faster.

Be sure to note whether your graph had all these characteristics, and whether your description included these characteristics. Note also any characteristics included in your description that were not included here.

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I thought the graph of y=2^x had a horizontal asymptote at x=0 since it never really reaches 0.

At this point we're only looking at what can be concluded from x values -3, -2, ..., 3. However the pattern that is established here is, as you already understand, easily extended to see why y = 2^x has the horizontal asymptote at the negative x axis, and also why it 'blows up' rapidly, though not with an asymptote, for positive x.

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00:31:37

`q006. Make a table for y = x^2 + 3, using x values -3, -2, -1, 0, 1, 2, 3. How do the y values on the table compare to the y values on the table for y = x^2? How does the graph of y = x^2 + 3 compare to the graph of y = x^2?

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the y values are 12, 7, 4, 3, 4, 7, 12

you added 3 to the values so each value is increased by 3 from the y=x^2 equation

The graph is moved 3 units upward since it is plus 3 minus 3 would be downward.

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00:31:49

A list of the y values will include, in order, y = 12, 7, 4, 3, 4, 7, 12.

A list for y = x^2 would include, in order, y = 9, 4, 1, 0, 1, 4, 9.

The values for y = x^2 + 3 are each 3 units greater than those for the function y = x^2.

The graph of y = x^2 + 3 therefore lies 3 units higher at each point than the graph of y = x^2.

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00:36:57

`q007. Make a table for y = (x -1)^3, using x values -3, -2, -1, 0, 1, 2, 3. How did the values on the table compare to the values on the table for y = x^3? Describe the relationship between the graph of y = (x -1)^3 and y = x^3.

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y values are -64, -27, -8, -1, 0, 1, 8

The values made the graph stretch and made the point of symmetry (0,-1)

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00:37:34

The values you obtained should have been -64, -27, -8, -1, 0, 1, 8.

The values for y = x^3 are -27, -8, -1, 0, 1, 8, 27.

The values of y = (x-1)^3 are shifted 1 position to the right relative to the values of y = x^3. The graph of y = (x-1)^3 is similarly shifted 1 unit to the right of the graph of y = x^3.

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00:42:08

`q008. Make a table for y = 3 * 2^x, using x values -3, -2, -1, 0, 1, 2, 3. How do the values on the table compare to the values on the table for y = 2^x? Describe the relationship between the graph of y = 3 * 2^x and y = 2^x.

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y values are 3/8, 3/4, 1.5, 3, 6, 12, 24

The values are multiplied by 3. The graph increases at a greater rate than the graph of y=2^x

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00:42:18

You should have obtained y values 3/8, 3/4, 3/2, 3, 6, 12 and 24.

Comparing these with the values 1/8, 1/4, 1/2, 1, 2, 4, 8 of the function y = 2^x we see that the values are each 3 times as great.

The graph of y = 3 * 2^x has an overall shape similar to that of y = 2^x, but each point lies 3 times as far from the x-axis. It is also worth noting that at every point the graph of y = 3 * 2^x is three times as the past that of y = 2^x.

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assignment 5

Your work on this assignment is very good. Let me know if you have questions.