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SAT II Math I : Properties of Functions and Graphs

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

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6 Diagnostic Tests 113 Practice Tests Question of the Day Flashcards Learn by Concept

Example Questions

Example Question #1 : Slope

What is the slope of the following equation? $\displaystyle 2y = 3-7x$

Possible Answers:

$\displaystyle 3$

$\displaystyle 14$

$\displaystyle -14$

$-\frac{7}{2}$ $\displaystyle -\frac{7}{2}$

$\frac{3}{2}$ $\displaystyle \frac{3}{2}$

Correct answer:

$-\frac{7}{2}$ $\displaystyle -\frac{7}{2}$

Explanation:

The given equation will need to be rewritten in slope intercept format.

y=mx+b $\displaystyle y=mx+b$

Divide by two on both sides.

$\frac{2y }{2}=\frac{ 3-7x}{2}$ $\displaystyle \frac{2y }{2}=\frac{ 3-7x}{2}$

Rearrange the right side by order of powers.

$y= -\frac{7}{2}x+\frac{3}{2}$ $\displaystyle y= -\frac{7}{2}x+\frac{3}{2}$

The slope can be seen as $-\frac{7}{2}$ $\displaystyle -\frac{7}{2}$

The answer is: $-\frac{7}{2}$ $\displaystyle -\frac{7}{2}$

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Example Question #41 : Functions And Graphs

What is the slope of the given equation? $y=-\frac{1}{2x}+3$ $\displaystyle y=-\frac{1}{2x}+3$

Possible Answers:

$\displaystyle 2$

$\displaystyle 3$

$\displaystyle -2$

$\textup{The slope does not exist.}$

$-\frac{1}{2}$ $\displaystyle -\frac{1}{2}$

Correct answer:

$\textup{The slope does not exist.}$

Explanation:

The slope in a linear equation is defined as y=mx+b $\displaystyle y=mx+b$ .

The x-variable exists in the denominator, which refers to the parent function of:

$y=\frac{1}{x}$ $\displaystyle y=\frac{1}{x}$

This function is not linear, and will have changing slope along its domain.

The answer is: $\textup{The slope does not exist.}$

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Example Question #1 : Maximum And Minimum

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

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

Possible Answers:

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

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

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

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

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

Correct answer:

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

Explanation:

The equation of a parabola can be written in vertex form

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

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

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

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

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

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

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Example Question #1 : Maximum And Minimum

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

Possible Answers:

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

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

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

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

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

Correct answer:

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

Explanation:

Rewrite the equation by the order of powers.

$\displaystyle y=-9x^2+6x+6$

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

Determine the values of a, b $\displaystyle a, b$ to the vertex formula.

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

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,

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

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All SAT II Math I Resources

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SAT II Math I : Properties of Functions and Graphs

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

All SAT II Math I Resources

Example Questions

Example Question #1 : Slope

Example Question #41 : Functions And Graphs

Example Question #1 : Maximum And Minimum

Example Question #1 : Maximum And Minimum

All SAT II Math I Resources

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