005. ********************************************* Question: `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? YYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYY Your solution: When we plug in the following x values: Y=(-3)^2 y= 9 Y=(-2)^2 y= 4 Y=(-1)^2 y= 1 Y=(0)^2 y= 0 Y=(1)^2 y= 1 Y=(2)^2 y= 4 Y=(3)^2 y= 9 confidence rating #$&*10 ^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^
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Given Solution: You should have obtained y values 9, 4, 1, 0, 1, 4, 9, in that order. &&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&& Self-critique (if necessary): ------------------------------------------------ Self-critique rating #$&* ********************************************* Question: `q002. Evaluate the function y = 2^x for x values -3, -2, -1, 0, 1, 2, and 3. What are your y values? YYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYY Your solution: When we plug in the following x values : Y=2^(-3) y= .125 or 1/8 Y=2^(-2) y= .25 or 1/4 Y=2^(-1) y= .5 or 1/2 Y=2^(0) y= 1 Y=2^(1) y= 2 Y=2^(2) y= 4 Y=2^(3) y= 8 confidence rating #$&*10 ^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^
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Given Solution: By the laws of exponents, b^-x = 1 / b^x. So for example 2^-2 = 1 / 2^2 = 1/4. Your y values will be 1/8, 1/4, 1/2, 1, 2, 4 and 8. Note that we have used the fact that for any b, b^0 = 1. It is a common error to say that 2^0 is 0. Note that this error would interfere with the pattern or progression of the y values. &&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&& Self-critique (if necessary): ------------------------------------------------ Self-critique rating #$&* ********************************************* Question: `q003. Evaluate the function y = x^-2 for x values -3, -2, -1, 0, 1, 2, and 3. What are your y values? YYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYY Your solution: When we plug in the following x values: Y= (-3)^-2 y= 1/9 Y=(-2)^-2 y= 1/4 Y=(-1)^-2 y= 1 Y= (0)^-2 y= undefined Y=(1)^-2 y= 1 Y=(2)^-2 y= 1/4 Y= (3)^-2 y= 1/9 confidence rating #$&* 10 ^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^
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Given Solution: By the laws of exponents, x^-p = 1 / x^p. So x^-2 = 1 / x^2, and your x values should be 1/9, 1/4, and 1. Since 1 / 0^2 = 1 / 0 and division by zero is not defined, the x = 0 value is undefined. The last three values will be 1, 1/4, and 1/9. &&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&& Self-critique (if necessary): ------------------------------------------------ Self-critique rating #$&* ********************************************* Question: `q004. Evaluate the function y = x^3 for x values -3, -2, -1, 0, 1, 2, and 3. What are your y values? YYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYY Your solution: When we plug in the following x values: Y= (-3)^3 y= -27 Y= (-2)^3 y= -8 Y= (-1)^3 y= -1 Y= (0)^3 y= 0 Y= (1)^3 y= 1 Y= (2)^3 y= 8 Y= (3)^3 y= 27 confidence rating #$&* 10 ^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^
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Given Solution: The y values should be -27, -8, -1, 0, 1, 8, 27 &&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&& Self-critique (if necessary): ------------------------------------------------ Self-critique rating #$&* ********************************************* Question: `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. YYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYY Your solution: The graph y=x^2 is a parabola with its vertex at the origin. The axis of symmetry is the y axis with each side being equal to each other. When x goes up by ones unit each time y doubles. The graph of y = 2^x begins at x = -3 when y= 1/8. With 1/8 being close to zero. The graph starts on the left , close to the x-axis since 3 is negative. As x moves to the right, the y value doubles. This causes the graph to rise more quickly as we move from left to right. The graph of y = x^-2 rises quickly 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. On the graph of y = x ^ 3 the x and y values are both positive or both negative. confidence rating #$&*3 ^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^
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Given Solution: The graph of y = x^2 is a parabola with its vertex at the origin. It is worth noting that the graph is symmetric with respect to the y-axis. That is, the graph to the left of the y-axis is a mirror image of the graph to the right of the y-axis. The graph of y = 2^x begins at x = -3 with value 1/8, which is relatively close to zero. The graph therefore starts to the left, close to the x-axis. With each succeeding unit of x, with x moving to the right, the y value doubles. This causes the graph to rise more and more quickly as we move from left to right. The graph intercepts the y-axis at y = 1. The graph of y = x^-2 rises more and more rapidly as we approach the y-axis from the left. It might not be clear from the values obtained here that this progression continues, with the y values increasing beyond bound, but this is the case. This behavior is mirrored on the other side of the y-axis, so that the graph rises as we approach the y-axis from either side. In fact the graph rises without bound as we approach the y-axis from either side. The y-axis is therefore a vertical asymptote for this graph. The graph of y = x ^ 3 has negative y values whenever x is negative and positive y values whenever x is positive. As we approach x = 0 from the left, through negative x values, the y values increase toward zero, but the rate of increase slows so that the graph actually levels off for an instant at the point (0,0) before beginning to increase again. To the right of x = 0 the graph increases faster and faster. Be sure to note whether your graph had all these characteristics, and whether your description included these characteristics. Note also any characteristics included in your description that were not included here. &&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&& Self-critique (if necessary): ------------------------------------------------ Self-critique rating #$&* ********************************************* Question: `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? YYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYY Your solution: X Y -3 12 -2 7 -1 4 0 3 1 4 2 7 3 12 The y values for y=x^2 were y= 9, 4, 1, 0, 1, 4, 9 The y values for y=x^2+3 are all 3 units more than those for y=x^2 because we added 3 to the original function. confidence rating #$&*7 ^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^
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Given Solution: A list of the y values will include, in order, y = 12, 7, 4, 3, 4, 7, 12. A list for y = x^2 would include, in order, y = 9, 4, 1, 0, 1, 4, 9. The values for y = x^2 + 3 are each 3 units greater than those for the function y = x^2. The graph of y = x^2 + 3 therefore lies 3 units higher at each point than the graph of y = x^2. &&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&& Self-critique (if necessary): ------------------------------------------------ Self-critique rating #$&* ********************************************* Question: `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. YYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYY Your solution: When we plug the following x values: Y=(-3 -1)^3 y= -64 Y= (-2-1)^3 y= -27 Y=(-1-1)^3 y= -8 Y=(0-1)^3 y= -1 Y=(1-1)^3 y= 0 Y=(2-1)^3 y= 1 Y=(3-1)^3 y= 8 The values for y = x^3 are -27, -8, -1, 0, 1, 8, 27. Because of subtracting 1 from the function y=(x-1)^3 this shifted the graph 1 unit to the right of the graph y= x^3. confidence rating #$&*6 ^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^
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Given Solution: The values you obtained should have been -64, -27, -8, -1, 0, 1, 8. The values for y = x^3 are -27, -8, -1, 0, 1, 8, 27. The values of y = (x-1)^3 are shifted 1 position to the right relative to the values of y = x^3. The graph of y = (x-1)^3 is similarly shifted 1 unit to the right of the graph of y = x^3. STUDENT QUESTION I assumed the graph was shifted 1 unit down since the graph passes through (0, -1) instead of origin. Then again, it passes through (1, 0), so could it be said that the graph is shifted 1 unit down OR 1 unit to the right? INSTRUCTOR RESPONSE Based on those two points that would be correct. Nowever, for example, (-2, -8) shifts to (-1, -8), a shift to the right, but not to (-2, -9), as would be the case if this was a downward shift. &&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&& Self-critique (if necessary): ------------------------------------------------ Self-critique rating #$&* ********************************************* Question: `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. YYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYY Your solution: When we plug in the following x values : Y= 3*2^(-3) y=3/8 Y=3*2^(-2) y=3/4 Y=3*2^(-1) y=3/2 Y=3*2^(0) y= 3 Y=3*2^(1) y=6 Y=3*2^(2) y=12 Y=3*2^(3) y=24 Compared to the values of y=2^x because we multiplied y=3*2^x by a factor of 3 it is a given that the y values will be 3 times greater. The shape of the graph y = 3 * 2^x is the same to that of y = 2^x, but each point lies 3 times as far from the x-axis. confidence rating #$&*7 ^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^
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Given Solution: You should have obtained y values 3/8, 3/4, 3/2, 3, 6, 12 and 24. Comparing these with the values 1/8, 1/4, 1/2, 1, 2, 4, 8 of the function y = 2^x we see that the values are each 3 times as great. The graph of y = 3 * 2^x has an overall shape similar to that of y = 2^x, but each point lies 3 times as far from the x-axis. It is also worth noting that at every point the graph of y = 3 * 2^x is three times as the past that of y = 2^x. &&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&& Self-critique (if necessary): ------------------------------------------------ Self-critique rating #$&* "