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

$$f(x)=(x-14)(x+19)$$

The function $f$ is defined by the given equation. For what value of $x$ does $f(x)$ 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-54"}],"about":{"@type":"Thing","name":"Math, Advanced Math, Duplicate"},"educationalLevel":"intermediate","eduQuestionType":"Multiple choice","learningResourceType":"Practice problem","assesses":["Math","Advanced Math","Duplicate"],"name":"

$$f(x)=(x-14)(x+19)$$

The function $f$ is defined by the given equation. For what value of $x$ does $f(x)$ reach its minimum?

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

Choice D is correct. It's given that $f(x)=(x-14)(x+19)$, which can be rewritten as $f(x)=x^{2}+5x-266$. Since the coefficient of the $x^{2}$-term is positive, the graph of $y=f(x)$ in the $xy$-plane opens upward and reaches its minimum value at its vertex. The $x$-coordinate of the vertex is the value of $x$ such that $f(x)$ reaches its minimum. For an equation in the form $f(x)=ax^{2}+bx+c$, where $a,b,$ and $c$ are constants, the $x$-coordinate of the vertex is $-\\frac{b}{2a}$. For the equation $f(x)=x^{2}+5x-266$, $a=1,b=5$, and $c=-266$. It follows that the $x$-coordinate of the vertex is $-\\frac{5}{2(1)}$, or $-\\frac{5}{2}$. Therefore, $f(x)$ reaches its minimum when the value of $x$ is $-\\frac{5}{2}$.

Alternate approach: The value of $x$ for the vertex of a parabola is the $x$-value of the midpoint between the two $x$-intercepts of the parabola. Since it's given that $f(x)=(x-14)(x+19)$, it follows that the two $x$-intercepts of the graph of $y=f(x)$ in the $xy$-plane occur when $x=14$ and $x=-19$, or at the points $(14,0)$ and $(-19,0)$. The midpoint between two points, $(x_{1},y_{1})$ and $(x_{2},y_{2})$, is $\\left(\\frac{x_{1}+x_{2}}{2},\\,\\frac{y_{1}+y_{2}}{2}\\right)$. Therefore, the midpoint between $(14,0)$ and $(-19,0)$ is $\\left(\\frac{14+(-19)}{2},\\,\\frac{0+0}{2}\\right)$, or $\\left(-\\frac{5}{2},0\\right)$. It follows that $f(x)$ reaches its minimum when the value of $x$ is $-\\frac{5}{2}$

Choice A is incorrect. This is the y-coordinate of the y-intercept of the graph of $y=f(x)$ in the xy-plane.

Choice B is incorrect. This is one of the x-coordinates of the x-intercepts of the graph of $y=f(x)$ in the xy-plane.

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

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

$$f(x)=(x-14)(x+19)$$

The function $f$ is defined by the given equation. For what value of $x$ does $f(x)$ reach its minimum?

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

$-266$

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

$-19$

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

$-\\frac{33}{2}$

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

$-\\frac{5}{2}$

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

Choice A is incorrect. This is the y-coordinate of the y-intercept of the graph of $y=f(x)$ in the xy-plane.

Choice B is incorrect. This is one of the x-coordinates of the x-intercepts of the graph of $y=f(x)$ in the xy-plane.

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

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

Choice A is incorrect. This is the y-coordinate of the y-intercept of the graph of $y=f(x)$ in the xy-plane.

Choice B is incorrect. This is one of the x-coordinates of the x-intercepts of the graph of $y=f(x)$ in the xy-plane.

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

"}}},"url":"https://classnotes.xyz/mcqs/sat-math-advanced-math-multiple-choice-question-54"}

$$f(x)=(x-14)(x+19)$$The function $f$ is defined by the given equation. For what value of $x$ does $f(x)$ reach its minimum?

$$f(x)=(x-14)(x+19)$$The function $f$ is defined by the given equation. For what value of $x$ does $f(x)$ reach its minimum?

$$f(x)=(x-14)(x+19)$$The function $f$ is defined by the given equation. For what value of $x$ does $f(x)$ reach its minimum?

$$f(x)=(x-14)(x+19)$$

The function $f$ is defined by the given equation. For what value of $x$ does $f(x)$ reach its minimum?

$$f(x)=(x-14)(x+19)$$

The function $f$ is defined by the given equation. For what value of $x$ does $f(x)$ reach its minimum?

$$f(x)=(x-14)(x+19)$$

The function $f$ is defined by the given equation. For what value of $x$ does $f(x)$ reach its minimum?