# Assignment -with answer - 1 A graph has a point that is...

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1 1. A graph has a point that is either a maximum or a minimum. To the left of the point, the slope of relationship is positive. To the right of the point, the slope is negative. Is the point a maximum point or a minimum point? Be sure to draw a figure that supports your answer. Answer: The point is a maximum point. Examine the figure above. The slope of a curved line at any point equals the slope of a straight line that touches the curved line at only that one point. Thus to the left of the maximum point, take point A. The slope of the straight line that touches the curved line at only point A is positive, so the slope of the relationship is positive. Similarly, take point B to the right of the maximum point. As the straight line shows, the slope of the relationship at point B is negative. Indeed, whenever there is a maximum point, the slope of the relationship to the left of the maximum is positive and the slope to the right is negative. 2. If two points on a line are x = 2, y = 5 and x = 7, y = 10, what is the slope of this line? Answer: The slope equals the change in the y-variable divided by the change in the x- variable. So, the slope equals (10 - 5)/(7 - 2 ) = (5)/(5) = 1.00. 3. The table below shows how the number of books Katie buys each year depends on her income. Katie's income (dollars per year) Katie's purchases (books per year) 50,000 14 70,000 16 90,000 18 110,000 20 a) What kind of relationship exists between Katie's income and the number of books she purchases?

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2 b) Plot the relationship between Katie's income and the number of books she purchases. Measure income along the vertical axis and the number of books along the horizontal axis. c) What is the slope of the relationship between \$50,000 and \$70,000 of income? d) What is the slope of the relationship between \$90,000 and \$110,000 of income? e) Comment on the similarity or dissimilarity of your answers to parts (c) and (d). Answer: a) There is a positive relationship. When Katie's income increases, so too does her purchase of books. b) The relationship is plotted in the figure above. c) The slope equals the change in the value of the variable measured on the vertical axis, income, divided by the change in the value of the variable measured along the horizontal axis, the number of books. Between \$50,000 and \$70,000 of income, the number of books purchased increases from 14 to 16. Hence income increases by \$20,000 and the number of books increases by 2, so the slope equals \$20,000/2 = 10,000. d) As with the previous answer, the slope equals the change in income divided by the change in books. Between \$90,000 and \$110,000 of income, the number of books purchased increases from 18 to 20. Hence income increases by \$20,000 and the number of books increases by 2, so the slope equals \$20,000/2 = 10,000. e) The slopes in parts (c) and (d) are equal. But, they must be equal because the relationship between Katie's income and the number of books she purchases is linear. For a linear relationship, the slope is the same regardless of where it is measured.
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