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Precalculus : Graphing Functions

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Example Question #1 : Graphing Functions

Information about the Hardy Weinberg equation: http://www.nature.com/scitable/definition/hardy-weinberg-equation-299

The Hardy Weinberg equation is a very important concept in population genetics. Suppose we have two "alleles" for a specific trait (like eye color, gender, etc..) The proportion for which allele is present is given by $\displaystyle x$ and $\displaystyle y$

Then by Hardy-Weinberg:

$\displaystyle (x+y)^2 = 1$ , where both $\displaystyle x$ and $\displaystyle y$ are non-negative.

Which statement discribes the graph that would appropriately represents the above relation?

Possible Answers:

Curved line passing through the point $\displaystyle (0.6, 0.9)$

Straight line passing through the point $\displaystyle (0.5, 0.5)$

Parabola opening up passing through the point $\displaystyle (1, 0.5)$

Straight line passing through the point $\displaystyle (0.5, 1)$

Correct answer:

Straight line passing through the point $\displaystyle (0.5, 0.5)$

Explanation:

The equation can be reduced by taking the square root of both sides.

$\displaystyle (x+y)^2 = 1$

$\displaystyle (x+y) = 1$

$\displaystyle y = 1 - x$

As a simple test, all other values when substituted into the original equation fail. However, $\displaystyle (0.5 + 0.5)^2 = 1$ works. Therefore $\displaystyle (0.5,0.5)$ is our answer.

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Example Question #1 : Graphing Functions

What is the y-intercept of the following equation?

$f(x)=4\sin{3x}+2$ $\displaystyle f(x)=4\sin{3x}+2$

Possible Answers:

(0,2) $\displaystyle (0,2)$

(2,0) $\displaystyle (2,0)$

(0,4) $\displaystyle (0,4)$

(0,3) $\displaystyle (0,3)$

(4,0) $\displaystyle (4,0)$

Correct answer:

(0,2) $\displaystyle (0,2)$

Explanation:

The y-intercept can by found by solving the equation when x=0. Thus,

$f(0)=4\sin(3*0)+2$ $\displaystyle f(0)=4\sin(3*0)+2$

$\displaystyle f(0)=4*0+2$

f(0)=2 $\displaystyle f(0)=2$

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Example Question #2 : Graphing Functions

Determine the y intercept of f(x) $\displaystyle f(x)$, where $f(x)=\frac{x-4}{x^{2}+x-12}$ $\displaystyle f(x)=\frac{x-4}{x^{2}+x-12}$ .

Possible Answers:

$\bigg(\frac{1}{3},0\bigg)$ $\displaystyle \bigg(\frac{1}{3},0\bigg)$

(4,0) $\displaystyle (4,0)$

(12,0) $\displaystyle (12,0)$

(0,12) $\displaystyle (0,12)$

$\bigg(0,\frac{1}{3}\bigg)$ $\displaystyle \bigg(0,\frac{1}{3}\bigg)$

Correct answer:

$\bigg(0,\frac{1}{3}\bigg)$ $\displaystyle \bigg(0,\frac{1}{3}\bigg)$

Explanation:

In order to determine the y-intercept of f(x) $\displaystyle f(x)$, set x=0 $\displaystyle x=0$

$f(0)=\frac{0-4}{0^{2}+0-12}$ $\displaystyle f(0)=\frac{0-4}{0^{2}+0-12}$

$f(0)=\frac{-4}{-12}$ $\displaystyle f(0)=\frac{-4}{-12}$

$f(0)=\frac{1}{3}$ $\displaystyle f(0)=\frac{1}{3}$

Solving for y, when x is equal to zero provides you with the y coordinate for the intercept. Thus the y-intercept is $\bigg(0,\frac{1}{3}\bigg)$ $\displaystyle \bigg(0,\frac{1}{3}\bigg)$.

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Example Question #2 : Graphing Functions

What is the $\displaystyle y$-intercept of the function,

$\displaystyle f(x)=(x-2)^2+4$?

Possible Answers:

$\displaystyle 2$

$\displaystyle 0$

$\displaystyle -2$

$\displaystyle 8$

$\displaystyle 4$

Correct answer:

$\displaystyle 8$

Explanation:

To find the $\displaystyle y$-intercept we need to find the cooresponding $\displaystyle y$ value when x = 0 $\displaystyle x = 0$.

Substituting x = 0 $\displaystyle x = 0$ into our function we get the following:

$\displaystyle f(0)=(0-2)^2+4$

$\displaystyle f(0)=4+4$

f(0)=8 $\displaystyle f(0)=8$

Therefore, our $\displaystyle y$-intercept is $\displaystyle 8$.

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Example Question #1 : Graphing Functions

What is the value of the $\displaystyle y$-intercept of $\displaystyle f(x)=(2x+4)^2+5$?

Possible Answers:

$\displaystyle 11$

$\displaystyle 16$

$\displaystyle 5$

The graph does not have a $\displaystyle y$-intercept

$\displaystyle 21$

Correct answer:

$\displaystyle 21$

Explanation:

To find the $\displaystyle y$-intercept we need to find the cooresponding $\displaystyle y$ value when x=0 $\displaystyle x=0$. Therefore, we substitute in x=0 $\displaystyle x=0$ and solve:

$\displaystyle f(0)=(2(0)+4)^2+5$

$\displaystyle f(0)=(0+4)^2+5$

$\displaystyle f(0)=16+5$

f(0)=21 $\displaystyle f(0)=21$

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Example Question #3 : Graphing Functions

If $\displaystyle f(x) = f(-x)$, what kind of symmetry does the function f(x) $\displaystyle f(x)$ have?

Possible Answers:

Even Symmetry

No Symmetry

Odd Symmetry

Symmetry across the line y=x

Correct answer:

Even Symmetry

Explanation:

The definition of even symmetry is if $\displaystyle f(x) = f(-x)$

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Example Question #3 : Graphing Functions

If $\displaystyle f(-x)=-f(x)$, what kind of symmetry does f(x) $\displaystyle f(x)$ have?

Possible Answers:

Even symmetry

Symmetry across the line y=x

Odd symmetry

No symmetry

Correct answer:

Odd symmetry

Explanation:

$\displaystyle f(-x)=-f(x)$ is the definition of odd symmetry

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Example Question #4 : Graphing Functions

Solve for $\displaystyle n$.

(Figure not drawn to scale).

Possible Answers:

14^o $\displaystyle 14^o$

10^o $\displaystyle 10^o$

8^o $\displaystyle 8^o$

12^o $\displaystyle 12^o$

Correct answer:

14^o $\displaystyle 14^o$

Explanation:

The angles are supplementary, therefore, the sum of the angles must equal 180^o $\displaystyle 180^o$.

$\small (4n+22^o)+(8n-10^o)=180^o$ $\displaystyle \small (4n+22^o)+(8n-10^o)=180^o$

$\small 4n+22^o+8n-10^o=180^o$ $\displaystyle \small 4n+22^o+8n-10^o=180^o$

$\small 12n+12^o=180^o$ $\displaystyle \small 12n+12^o=180^o$

$\small 12n=168^o$ $\displaystyle \small 12n=168^o$

$\small n=14^o$ $\displaystyle \small n=14^o$

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Example Question #1 : Understanding Complementary And Suplmentary Angles

Are $129^{\circ}$ $\displaystyle 129^{\circ}$ and $51^{\circ}$ $\displaystyle 51^{\circ}$ supplementary angles?

Possible Answers:

Not enough information

Yes

Correct answer:

Yes

Explanation:

Since supplementary angles must add up to $180^{\circ}$ $\displaystyle 180^{\circ}$, the given angles are indeed supplementary.

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Example Question #5 : Graphing Functions

Solve for $\displaystyle x$ and $\displaystyle y$.

(Figure not drawn to scale).

Possible Answers:

$\small x=15^o;\ y=37.5^o$ $\displaystyle \small x=15^o;\ y=37.5^o$

$\small x=15^o;\ y=52.5^o$ $\displaystyle \small x=15^o;\ y=52.5^o$

$\small x=7.5^o;\ y=18.75^o$ $\displaystyle \small x=7.5^o;\ y=18.75^o$

$\small x=15^o;\ y=7.5^o$ $\displaystyle \small x=15^o;\ y=7.5^o$

Correct answer:

$\small x=15^o;\ y=37.5^o$ $\displaystyle \small x=15^o;\ y=37.5^o$

Explanation:

The angles containing the variable $\displaystyle x$ all reside along one line, therefore, their sum must be 180^o $\displaystyle 180^o$.

$\small 5x+4x+3x=180^o$ $\displaystyle \small 5x+4x+3x=180^o$

$\small 12x=180^o$ $\displaystyle \small 12x=180^o$

$\small x=15^o$ $\displaystyle \small x=15^o$

Because $\displaystyle 2y$ and $\displaystyle 5x$ are opposite angles, they must be equal.

$\small 2y=5x$ $\displaystyle \small 2y=5x$

$\small x=15^o$ $\displaystyle \small x=15^o$

$\small 2y=5(15^o)=75^o$ $\displaystyle \small 2y=5(15^o)=75^o$

$\small y=\frac{75^o}{2}=37.5^o$ $\displaystyle \small y=\frac{75^o}{2}=37.5^o$

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Precalculus : Graphing Functions

Study concepts, example questions & explanations for Precalculus

All Precalculus Resources

Example Questions

Example Question #1 : Graphing Functions

Example Question #1 : Graphing Functions

Example Question #2 : Graphing Functions

Example Question #2 : Graphing Functions

Example Question #1 : Graphing Functions

Example Question #3 : Graphing Functions

Example Question #3 : Graphing Functions

Example Question #4 : Graphing Functions

Example Question #1 : Understanding Complementary And Suplmentary Angles

Example Question #5 : Graphing Functions

All Precalculus Resources

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