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

$f(x)=4x^{2}+64x+262$

The function $g$ is defined by $g(x)=f(x+5)$. For what value of $x$ does $g(x)$ reach its minimum?

","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-32"}],"about":{"@type":"Thing","name":"Math, Advanced Math, Duplicate"},"educationalLevel":"intermediate","eduQuestionType":"Multiple choice","learningResourceType":"Practice problem","assesses":["Math","Advanced Math","Duplicate"],"name":"

$f(x)=4x^{2}+64x+262$

The function $g$ is defined by $g(x)=f(x+5)$. For what value of $x$ does $g(x)$ reach its minimum?

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

Choice A is correct. It's given that $g(x)=f(x+5)$. Since $f(x)=4x^{2}+64x+262$, it follows that $f(x+5)=4(x+5)^{2}+64(x+5)+262$. Expanding the quantity $(x+5)^{2}$ in this equation yields $f(x+5)=4(x^{2}+10x+25)+64(x+5)+262$. Distributing the $4$ and the $64$ yields $f(x+5)=4x^{2}+40x+100+64x+320+262$. Combining like terms yields $f(x+5)=4x^{2}+104x+682$. Therefore, $g(x)=4x^{2}+104x+682$. For a quadratic function defined by an equation of the form $g(x)=a(x-h)^{2}+k$, where $a$, $h$, and $k$ are constants and $a$ is positive, $g(x)$ reaches its minimum, $k$, when the value of $x$ is $h$. The equation $g(x)=4x^{2}+104x+682$ can be rewritten in this form by completing the square. This equation is equivalent to $g(x)=4(x^{2}+26)+682$, or $g(x)=4(x^{2}+26x+169-169)+682$. This equation can be rewritten as $g(x)=4[(x+13)^{2}-169]+682$, or $g(x)=4(x+13)^{2}-4(169)+682$, which is equivalent to $g(x)=4(x+13)^{2}+6$. This equation is in the form $g(x)=a(x-h)^{2}+k$, where $a=4$, $h=-13$, and $k=6$. Therefore, $g(x)$ reaches its minimum when the value of $x$ is $-13$.

Choice B is incorrect. This is the value of $x$ for which $f(x)$, rather than $g(x)$, reaches its minimum.

Choice C is incorrect and may result from conceptual or calculation errors.

Choice D is incorrect. This is the value of $x$ for which $f(x-5)$, rather than $f(x+5)$, reaches its minimum.

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

$f(x)=4x^{2}+64x+262$

The function $g$ is defined by $g(x)=f(x+5)$. For what value of $x$ does $g(x)$ reach its minimum?

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

$-8$

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

$-5$

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

$-3$

"}],"acceptedAnswer":{"@type":"Answer","position":0,"encodingFormat":"text/html","text":"

$-13$

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

Choice B is incorrect. This is the value of $x$ for which $f(x)$, rather than $g(x)$, reaches its minimum.

Choice C is incorrect and may result from conceptual or calculation errors.

Choice D is incorrect. This is the value of $x$ for which $f(x-5)$, rather than $f(x+5)$, reaches its minimum.

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

Choice B is incorrect. This is the value of $x$ for which $f(x)$, rather than $g(x)$, reaches its minimum.

Choice C is incorrect and may result from conceptual or calculation errors.

Choice D is incorrect. This is the value of $x$ for which $f(x-5)$, rather than $f(x+5)$, reaches its minimum.

"}}},"url":"https://classnotes.xyz/mcqs/sat-math-advanced-math-multiple-choice-question-32"}

$f(x)=4x^{2}+64x+262$The function $g$ is defined by $g(x)=f(x+5)$. For what value of $x$ does $g(x)$ reach its minimum?

$f(x)=4x^{2}+64x+262$The function $g$ is defined by $g(x)=f(x+5)$. For what value of $x$ does $g(x)$ reach its minimum?

$f(x)=4x^{2}+64x+262$The function $g$ is defined by $g(x)=f(x+5)$. For what value of $x$ does $g(x)$ reach its minimum?

$f(x)=4x^{2}+64x+262$

The function $g$ is defined by $g(x)=f(x+5)$. For what value of $x$ does $g(x)$ reach its minimum?

$f(x)=4x^{2}+64x+262$

The function $g$ is defined by $g(x)=f(x+5)$. For what value of $x$ does $g(x)$ reach its minimum?

$f(x)=4x^{2}+64x+262$

The function $g$ is defined by $g(x)=f(x+5)$. For what value of $x$ does $g(x)$ reach its minimum?