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$f(x)=(x+4)(x-1)(2x-3)$

The function $f$ is defined above. Which of the following is NOT an $x$-intercept of the graph of the function in the $xy$-plane?

","hasPart":{"@type":"Question","typicalAgeRange":"12-22","educationalAlignment":[{"@type":"AlignmentObject","alignmentType":"educationalSubject","targetName":"SAT"},{"@type":"AlignmentObject","alignmentType":"educationalLevel","targetName":"SAT","educationalFramework":"Common Core","targetUrl":"https://classnotes.xyz/mcqs/sat-math-advanced-math-multiple-choice-question-185"}],"about":{"@type":"Thing","name":"Math, Advanced Math, Duplicate"},"educationalLevel":"intermediate","eduQuestionType":"Multiple choice","learningResourceType":"Practice problem","assesses":["Math","Advanced Math","Duplicate"],"name":"

$f(x)=(x+4)(x-1)(2x-3)$

The function $f$ is defined above. Which of the following is NOT an $x$-intercept of the graph of the function in the $xy$-plane?

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

Choice B is correct. The graph of the function $f$ in the $xy$-plane has $x$-intercepts at the points $(x,y)$, where $y=f(x)=0$. Substituting $0$ for $f(x)$ in the given equation yields $0=(x+4)(x-1)(2x-3)$. By the zero product property, if $0=(x+4)(x-1)(2x-3)$, then $x+4=0$, $x-1=0$, or $2x-3=0$. Solving each of these linear equations for $x$, it follows that $x=-4$, $x=1$, and $x=\\dfrac{3}{2}$, respectively. This means that the graph of the function $f$ in the $xy$-plane has three $x$-intercepts: $ (-4,0) $, $ (1,0) $, and $ \\left(\\dfrac{3}{2},0\\right) $. Therefore, $ \\left(-\\dfrac{2}{3},0\\right) $ isn't an $x$-intercept of the graph of the function $f$.

Alternate approach: Substitution may be used. Since by definition an $x$-intercept of any graph is a point in the form $(k,0)$ where $k$ is a constant, and since all points in the options are in this form, it need only be checked whether the points in the options lie on the graph of the function $f$. Substituting $-\\frac{2}{3}$ for $x$ and $0$ for $f(x)$ in the given equation yields $0=\\left(-\\frac{2}{3}+4\\right)\\left(-\\frac{2}{3}-1\\right)\\left(2\\left(-\\frac {2}{3}\\right)-3\\right)$, or $0=\\frac{650}{27}$. Therefore, the point $ \\left(-\\frac{2}{3},0\\right) $ doesn't lie on the graph of the function $f$ and can't be an $x$-intercept of the graph.

Choices A, C, and D are incorrect because each of these points is an $x$-intercept of the graph of the function $f$ in the $xy$-plane. By definition, an $x$-intercept is a point on the graph of the form $(k,0)$, where $k$ is a constant. Substituting $-4$ for $x$ and $0$ for $f(x)$ in the given equation yields $0=(-4+4)(-4-1)(2(-4)-3)$, or $0=0$. Since this is a true statement, the point $ (-4,0) $ lies on the graph of the function $f$ and is an $x$-intercept of the graph. Performing similar substitution using the points $ (1,0) $ and $ \\left(\\dfrac{3}{2},0\\right) $ also yields the true statement $0=0$, illustrating that these points also lie on the graph of the function $f$ and are $x$-intercepts of the graph.

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$f(x)=(x+4)(x-1)(2x-3)$

The function $f$ is defined above. Which of the following is NOT an $x$-intercept of the graph of the function in the $xy$-plane?

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

$ (-4,0) $

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

$ (1,0) $

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

$ \\left(\\frac{3}{2},0\\right) $

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

$ \\left(-\\frac{2}{3},0\\right) $

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

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"}}},"url":"https://classnotes.xyz/mcqs/sat-math-advanced-math-multiple-choice-question-185"}

$f(x)=(x+4)(x-1)(2x-3)$The function $f$ is defined above. Which of the following is NOT an $x$-intercept of the graph of the function in the $xy$-plane?

$f(x)=(x+4)(x-1)(2x-3)$The function $f$ is defined above. Which of the following is NOT an $x$-intercept of the graph of the function in the $xy$-plane?

$f(x)=(x+4)(x-1)(2x-3)$The function $f$ is defined above. Which of the following is NOT an $x$-intercept of the graph of the function in the $xy$-plane?

$f(x)=(x+4)(x-1)(2x-3)$

The function $f$ is defined above. Which of the following is NOT an $x$-intercept of the graph of the function in the $xy$-plane?

$f(x)=(x+4)(x-1)(2x-3)$

The function $f$ is defined above. Which of the following is NOT an $x$-intercept of the graph of the function in the $xy$-plane?

$f(x)=(x+4)(x-1)(2x-3)$

The function $f$ is defined above. Which of the following is NOT an $x$-intercept of the graph of the function in the $xy$-plane?