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SAT II Math I : Maximum and Minimum

Study concepts, example questions & explanations for SAT II Math I

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Example Questions

Example Question #1 : Maximum And Minimum

What is the vertex of the following function? Is it a maximum or a minimum?

$\small f(x) = -3(x-2)^2+9$

Possible Answers:

$\small \small (2,9); maximum$

$\small (2,-9); maximum$

$\small (2,9); minimum$

$\small (-2,9); maximum$

$\small (-2,9); minimum$

Correct answer:

$\small \small (2,9); maximum$

Explanation:

The equation of a parabola can be written in vertex form

$\small f(x) = a(x-h)^2+k$

where $\small \tiny (h,k)$ is the vertex and $\tiny a$ determines if it is a minimum or maximum. If $\tiny a$ is positive, then it is a minimum; if $\tiny a$ is negative, then it is a maximum.

$\small \small f(x)=-3(x-2)^2+9$

In this example, $\small \small a$ is negative, so the vertex is a maximum.

$\small h = 2$ and $\small k=9$

$\small (h,k)=(2,9)$

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Example Question #2 : Maximum And Minimum

Determine the maximum or minimum of y=6-9x^2+6x .

Possible Answers:

$\textup{Maximum at }x=-\frac{1}{3}$

$\textup{Maximum at }x=-3$

$\textup{Minimum at }x=\frac{2}{3}$

$\textup{Minimum at }x=-\frac{1}{3}$

$\textup{Maximum at }x=\frac{1}{3}$

Correct answer:

$\textup{Maximum at }x=\frac{1}{3}$

Explanation:

Rewrite the equation by the order of powers.

y=-9x^2+6x+6

This is a parabola in standard form: y=ax^2+bx+c

Determine the values of a, b to the vertex formula.

$x=-\frac{b}{2a}=-\frac{6}{2(-9)}= \frac{6}{18}=\frac{1}{3}$

Since the leading coefficient of the parabola is negative, the parabola will curve downward and will have a maximum point.

Therefore there is a maximum at,

$x=\frac{1}{3}$ .

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SAT II Math I : Maximum and Minimum

Study concepts, example questions & explanations for SAT II Math I

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Example Question #1 : Maximum And Minimum

Example Question #2 : Maximum And Minimum

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