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Advanced Math Difficulty: Hard

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 $x y$ -plane passes through the points ( $0$ , $10$ ) and ( $-2$ , $\frac{325}{36}$ ). What is the value of $a b$ ?

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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 $left parenthesis 0 comma 10 right parenthesis$ and $(- 2, \frac{325}{36})$ . 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 $9 equals b$ . Substituting $negative 2$ for $x$ and $StartFraction 325 Over 36 EndFraction$ for $h (x)$ in the equation $h (x) = a^{x} + 9$ yields $\frac{325}{36} = a^{- 2} + 9$ . Subtracting $9$ from both sides of this equation yields $\frac{1}{36} = a^{- 2}$ , which can be rewritten as $a squared equals 36$ . Taking the square root of both sides of this equation yields $a equals 6$ 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 $a b$ is $left parenthesis 6 right parenthesis left parenthesis 9 right parenthesis$ , or $54$ .

Choice A is incorrect and may result from finding the value of $a Superscript negative 2 Baseline b$ rather than the value of $a b$ .

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 $b$ .