{"@context":"https://schema.org/","@type":"Quiz","name":"\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n

$x$	$f(x)$
$-1$	$10$
$0$	$14$
$1$	$20$

For the quadratic function $f$, the table shows three values of $x$ and their corresponding values of $f(x)$. Which equation defines $f$?

","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-89"}],"about":{"@type":"Thing","name":"Math, Advanced Math, Duplicate"},"educationalLevel":"intermediate","eduQuestionType":"Multiple choice","learningResourceType":"Practice problem","assesses":["Math","Advanced Math","Duplicate"],"name":"\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n

$x$	$f(x)$
$-1$	$10$
$0$	$14$
$1$	$20$

For the quadratic function $f$, the table shows three values of $x$ and their corresponding values of $f(x)$. Which equation defines $f$?

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

Choice D is correct. The equation of a quadratic function can be written in the form $f(x)=a(x-h)^{2}+k$, where $a$, $h$, and $k$ are constants. It's given in the table that when $x=-1$, the corresponding value of $f(x)$ is $10$. Substituting $-1$ for $x$ and $10$ for $f(x)$ in the equation $f(x)=a(x-h)^{2}+k$ gives $10=a(-1-h)^{2}+k$, which is equivalent to $10=a(1+2h+h^{2})+k$, or $10=a+2ah+ah^{2}+k$. It's given in the table that when $x=0$, the corresponding value of $f(x)$ is $14$. Substituting $0$ for $x$ and $14$ for $f(x)$ in the equation $f(x)=a(x-h)^{2}+k$ gives $14=a(0-h)^{2}+k$, or $14=ah^{2}+k$. It's given in the table that when $x=1$, the corresponding value of $f(x)$ is $20$. Substituting $1$ for $x$ and $20$ for $f(x)$ in the equation $f(x)=a(x-h)^{2}+k$ gives $20=a(1-h)^{2}+k$, which is equivalent to $20=a(1-2h+h^{2})+k$, or $20=a-2ah+ah^{2}+k$. Adding $20=a-2ah+ah^{2}+k$ to the equation $10=a+2ah+ah^{2}+k$ gives $30=2a+2ah^{2}+2k$. Dividing both sides of this equation by $2$ gives $15=a+ah^{2}+k$. Since $14=ah^{2}+k$, substituting $14$ for $ah^{2}+k$ into the equation $15=a+4h^{2}+k$ gives $15=a+14$. Subtracting $14$ from both sides of this equation gives $a=1$. Substituting $1$ for $a$ in the equations $14=ah^{2}+k$ and $20=ah^{2}-2ah+a+k$ gives $14=h^{2}+k$ and $20=1-2h+h^{2}+k$, respectively. Since $14=h^{2}+k$, substituting $14$ for $h^{2}+k$ in the equation $20=1-2h+h^{2}+k$ gives $20=1-2h+14$, or $20=15-2h$. Subtracting $15$ from both sides of this equation gives $5=-2h$. Dividing both sides of this equation by $-2$ gives $-\\frac{5}{2}=h$. Substituting $-\\frac{5}{2}$ for $h$ into the equation $14=h^{2}+k$ gives $14=\\left(-\\frac{5}{2}\\right)^{2}+k$, or $14=\\frac{25}{4}+k$. Subtracting $\\frac{25}{4}$ from both sides of this equation gives $\\frac{31}{4}=k$. Substituting $1$ for $a$, $-\\frac{5}{2}$ for $h$, and $\\frac{31}{4}$ for $k$ in the equation $f(x)=a(x-h)^{2}+k$ gives $f(x)=\\left(x+\\frac{5}{2}\\right)^{2}+\\frac{31}{4}$, which is equivalent to $f(x)=x^{2}+5x+\\frac{25}{4}+\\frac{31}{4}$, or $f(x)=x^{2}+5x+14$. Therefore, $f(x)=x^{2}+5x+14$ defines $f$.

Choice A is incorrect. If $f(x)=3x^{2}+3x+14$, then when $x=-1$, the corresponding value of $f(x)$ is $14$, not $10$.

Choice B is incorrect. If $f(x)=5x^{2}+x+14$, then when $x=-1$, the corresponding value of $f(x)$ is $18$, not $10$.

Choice C is incorrect. If $f(x)=9x^{2}-x+14$, then when $x=-1$, the corresponding value of $f(x)$ is $24$, not $10$, and when $x=1$, the corresponding value of $f(x)$ is $22$, not $20$.

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

$x$	$f(x)$
$-1$	$10$
$0$	$14$
$1$	$20$

For the quadratic function $f$, the table shows three values of $x$ and their corresponding values of $f(x)$. Which equation defines $f$?

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

$f(x)=3x^{2}+3x+14$

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

$f(x)=5x^{2}+x+14$

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

$f(x)=9x^{2}-x+14$

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

$f(x)=x^{2}+5x+14$

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

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

"}}},"url":"https://classnotes.xyz/mcqs/sat-math-advanced-math-multiple-choice-question-89"}