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

In the $xy$-plane, the graph of the equation $y=-x^{2}+9x-100$ intersects the line $y=c$ at exactly one point. What is the value of $c$?

","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-204"}],"about":{"@type":"Thing","name":"Math, Advanced Math, Duplicate"},"educationalLevel":"intermediate","eduQuestionType":"Multiple choice","learningResourceType":"Practice problem","assesses":["Math","Advanced Math","Duplicate"],"name":"

In the $xy$-plane, the graph of the equation $y=-x^{2}+9x-100$ intersects the line $y=c$ at exactly one point. What is the value of $c$?

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

Choice C is correct. In the $xy$-plane, the graph of the line $y=c$ is a horizontal line that crosses the $y$-axis at $y=c$ and the graph of the quadratic equation $y=-x^{2}+9x-100$ is a parabola. A parabola can intersect a horizontal line at exactly one point only at its vertex. Therefore, the value of $c$ should be equal to the $y$-coordinate of the vertex of the graph of the given equation. For a quadratic equation in vertex form, $y=a(x-h)^{2}+k$, the vertex of its graph in the $xy$-plane is $\\left(h,k\\right)$. The given quadratic equation, $y=-x^{2}+9x-100$, can be rewritten as

$y=-\\left(x^{2}-2\\left(\\frac{9}{2}\\right)x+\\left(\\frac{9}{2}\\right)^{2}\\right) +\\left(\\frac{9}{2}\\right)^{2}-100$, or $y=-\\left(x-\\frac{9}{2}\\right)^{2}+\\left(-\\frac{319}{4}\\right)$. Thus, the value of $c$ is equal to $-\\frac{319}{4}$.

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

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

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

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

In the $xy$-plane, the graph of the equation $y=-x^{2}+9x-100$ intersects the line $y=c$ at exactly one point. What is the value of $c$?

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

$-\\frac{481}{4}$

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

$-100$

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

$-\\frac{9}{2}$

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

\$-\\frac{319}{4}\$

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

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

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

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

"}}},"url":"https://classnotes.xyz/mcqs/sat-math-advanced-math-multiple-choice-question-204"}