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The graph of the function $f$ is a line in the $xy$-plane. If the line has slope $\\frac{3}{4}$ and $f(0)=3$, which of the following defines $f$?

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The graph of the function $f$ is a line in the $xy$-plane. If the line has slope $\\frac{3}{4}$ and $f(0)=3$, which of the following defines $f$?

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

Choice B is correct. The equation for the function $f$ in the $xy$-plane can be represented by $f(x)=mx+b$, where $m$ is the slope and $b$ is the $y$-coordinate of the $y$-intercept. Since it's given that the line has a slope of $\\frac{3}{4}$, it follows that $m=\\frac{3}{4}$ in $f(x)=mx+b$, which yields $y=\\frac{3}{4}x+b$. It's given that $f(0)=3$. This implies that the graph of the function $f$ in the $xy$-plane passes through the point $(0,3)$. Thus, the y-coordinate of the $y$-intercept of the graph is 3, so $b=3$ in $f(x)=\\frac{3}{4}x+b$, which yields $f(x)=\\frac{3}{4}x+3$. Therefore, the equation $f(x)=\\frac{3}{4}x+3$ defines the function $f$.

Choice A is incorrect and may result from a sign error for the $y$-intercept. Choice C is incorrect and may result from using the denominator of the given slope as $m$ in $f(x)=mx+b$, in addition to a sign error for the $y$-intercept.

Choice D is incorrect and may result from using the denominator of the given slope as $m$ in $f(x)=mx+b$.

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The graph of the function $f$ is a line in the $xy$-plane. If the line has slope $\\frac{3}{4}$ and $f(0)=3$, which of the following defines $f$?

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

$f(x)=\\dfrac{3}{4}x-3$

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

$f(x)=4x-3$

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

$f(x)=4x+3$

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

$f(x)=\\dfrac{3}{4}x+3$

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"}}},"url":"https://classnotes.xyz/mcqs/sat-math-algebra-multiple-choice-question-210"}

The graph of the function $f$ is a line in the $xy$-plane. If the line has slope $\frac{3}{4}$ and $f(0)=3$, which of the following defines $f$?