The function h is defined by h(x) = a x + b, where a and b are positive constants. The graph of y = h(x) in the xy-plane passes through the points (0, 10) and (−2, 325 ⁄ 36 ). What is the value of ab?

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Key Explanation: Choice C is correct. It’s given that the function h is defined by h(x) = a x + b and that the graph of y = h(x) in the xy-plane passes through the points (0, 10) and (−2, 325 ⁄ 36 ). Substituting 0 for x and 10 for h(x) in the equation h(x) = a x + b yields 10 = a 0 + b, or 10 = 1 + b. Subtracting 1 from both sides of this equation yields 9 = b. Substituting −2 for x and 325 ⁄ 36 for h(x) in the equation h(x) = a x + 9 yields 325 ⁄ 36 = a −2 + 9. Subtracting 9 from both sides of this equation yields 1 ⁄ 36 = a −2 , which can be rewritten as 1 ⁄ 36 = 1 ⁄ a 2 , or a 2 = 36. Taking the square root of both sides of this equation yields a = 6 and a = −6, but because it’s given that a is a positive constant, a must equal 6. Because the value of a is 6 and the value of b is 9, the value of ab is (6)(9), or 54. Distractor Explanations: Choice A is incorrect and may result from finding the value of a −2 b rather than the value of ab. Choice B is incorrect and may result from conceptual or calculation errors. Choice D is incorrect and may result from correctly finding the value of a as 6, but multiplying it by the y-value in the first ordered pair rather than by the value of b.

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