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

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

","comment":{"@type":"Comment","text":"

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 $\\left(-2,\\frac{325}{36}\\right)$. 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 $\\frac{325}{36}$ 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 $a^{2}=36$. 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 $54$.

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 $b$.

"},"encodingFormat":"text/markdown","text":"

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

","suggestedAnswer":[{"@type":"Answer","position":0,"encodingFormat":"text/html","text":"

$\\frac{1}{4}$

"},{"@type":"Answer","position":1,"encodingFormat":"text/html","text":"

$\\frac{1}{2}$

"},{"@type":"Answer","position":3,"encodingFormat":"text/html","text":"

$60$

"}],"acceptedAnswer":{"@type":"Answer","position":2,"encodingFormat":"text/html","text":"

$54$

","comment":{"@type":"Comment","text":"

"},"answerExplanation":{"@type":"Comment","text":"

"}}},"url":"https://classnotes.xyz/mcqs/sat-math-advanced-math-multiple-choice-question-146"}

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