How to graph inverse variation - Geometry

Card 0 of 36

Question

A triangle is made up of the following points:

$\left { (2,0), (-3,5), (0,4) \right }$

What are the points of the inverse triangle?

Answer

The inverse of a function has all the same points as the original function, except the x values and y values are reversed. The same rule applies to polygons such as triangles.

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Question

Give the -intercept of the graph of the equation $y = \frac{4}{2x+ 5}$ .

Answer

Set in the equation:

$y = \frac{4}{2x+ 5}$

$y = \frac{4}{2 \cdot 0+ 5} = \frac{4}{ 5}$

The -intercept is $\left (0, \frac{4}{5} \right )$ .

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Question

Give the -intercept(s), if any, of the graph of the equation

$y = \frac{4}{2x+ 5}$

Answer

Set in the equation and solve for .

$y = \frac{4}{2x+ 5}$

$\frac{4}{2x+ 5} = 0$

$\frac{4}{2x+ 5} \cdot (2x+5) = 0 \cdot (2x+5)$

This is impossible, so the equation has no solution. Therefore, the graph has no -intercept.

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Question

Give the vertical asymptote of the graph of the equation

$y = \frac{2x-1}{3x+ 5}$

Answer

The vertical asymptote is , where is found by setting the denominator equal to 0 and solving for :

$3x \div 3 = - 5 \div 3$

$x= -\frac{5}{3}$

This is the equation of the vertical asymptote.

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Question

Give the -intercept(s), if any, of the graph of the equation

$y = \frac{2x-7}{3x+5}$

Answer

Set in the equation and solve for .

$y = \frac{2x-7}{3x+5}$

$0 = \frac{2x-7}{3x+5}$

$0 = \frac{2x-7}{3x+5} \cdot (3x+5)$

$2x \div 2 = 7 \div 2$

$x = \frac{7}{2}$

The -intercept is $\left ( \frac{7}{2}, 0 \right )$

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Question

Give the horizontal asymptote, if there is one, of the graph of the equation

$y = \frac{2x-7}{3x+5}$

Answer

To find the horizontal asymptote, we can divide both numerator and denominator in the right expression by :

$y = \frac{2x-7}{3x+5}$

$y = \frac{\frac{2x-7}{x}}{\frac{3x+5}{x}}$

$y = \frac{2- \frac{7}{x}}{3+ \frac{5}{x}}$

As approaches positive or negative infinity, $\frac{7}{x}$ and $\frac{5}{x}$ both approach 0. Therefore, approaches $\frac{2 }{3 }$ , making the horizontal asymptote the line of the equation $y = \frac{2 }{3 }$ .

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Question

Electrical power can be generated by wind, and the magnitude of power will depend on the wind speed. A wind speed of (in $\tiny m/s$ ) will generate a power of $v^{2}+4v$ . What is the minimum wind speed needed in order to power a device that requires ?

Answer

The simplest way to solve this problem is to plug all of the answer choices into the provided equation, and see which one results in a power of .

Alternatively, one could set up the equation,

$v{^2}+4v-96=0$ and factor, use the quadratic equation, or graph this on a calculator to find the root.

If we were to factor we would look for factors of c that when added together give us the value in b when we are in the form,

.

In our case . So we need factors of that when added together give us .

Thus the following factoring would solve this problem.

Then set each binomial equal to zero and solve for v.

$v+12=0 \rightarrow v=-12$

$v-8=0 \rightarrow v=8$

Since we can't have a negative power our answer is .

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Question

Compared to the graph $y=x^{2}-5$ , the graph $y=(x+4)^{2}-5$ has been shifted:

Answer

The inside the argument has the effect of shifting the graph units to the left. This can be easily seen by graphing both the original and modified functions on a graphing calculator.

Compare your answer with the correct one above

Question

If you look at and $y_{1}=(x-10)^2+10$ on the same graph. What is the transformation that took place from $y\rightarrow y_{1}$ ?

Answer

The graph shifts ten units to the right. The number inside the parentheses shows where and how many units the graph will shift. If it is in the parentheses with the x-coordinate, then it will shift either to the left or right. If the number is negative, it goes to the right. If it is positive, it will shift to the left.

If the number was with the y-coordinate then the graph would shift up and down.

Compare your answer with the correct one above

Question

A triangle is made up of the following points:

$\left { (2,0), (-3,5), (0,4) \right }$

What are the points of the inverse triangle?

Answer

The inverse of a function has all the same points as the original function, except the x values and y values are reversed. The same rule applies to polygons such as triangles.

Compare your answer with the correct one above

Question

Give the -intercept of the graph of the equation $y = \frac{4}{2x+ 5}$ .

Answer

Set in the equation:

$y = \frac{4}{2x+ 5}$

$y = \frac{4}{2 \cdot 0+ 5} = \frac{4}{ 5}$

The -intercept is $\left (0, \frac{4}{5} \right )$ .

Compare your answer with the correct one above

Question

Give the -intercept(s), if any, of the graph of the equation

$y = \frac{4}{2x+ 5}$

Answer

Set in the equation and solve for .

$y = \frac{4}{2x+ 5}$

$\frac{4}{2x+ 5} = 0$

$\frac{4}{2x+ 5} \cdot (2x+5) = 0 \cdot (2x+5)$

This is impossible, so the equation has no solution. Therefore, the graph has no -intercept.

Compare your answer with the correct one above

Question

Give the vertical asymptote of the graph of the equation

$y = \frac{2x-1}{3x+ 5}$

Answer

The vertical asymptote is , where is found by setting the denominator equal to 0 and solving for :

$3x \div 3 = - 5 \div 3$

$x= -\frac{5}{3}$

This is the equation of the vertical asymptote.

Compare your answer with the correct one above

Question

Give the -intercept(s), if any, of the graph of the equation

$y = \frac{2x-7}{3x+5}$

Answer

Set in the equation and solve for .

$y = \frac{2x-7}{3x+5}$

$0 = \frac{2x-7}{3x+5}$

$0 = \frac{2x-7}{3x+5} \cdot (3x+5)$

$2x \div 2 = 7 \div 2$

$x = \frac{7}{2}$

The -intercept is $\left ( \frac{7}{2}, 0 \right )$

Compare your answer with the correct one above

Question

Give the horizontal asymptote, if there is one, of the graph of the equation

$y = \frac{2x-7}{3x+5}$

Answer

To find the horizontal asymptote, we can divide both numerator and denominator in the right expression by :

$y = \frac{2x-7}{3x+5}$

$y = \frac{\frac{2x-7}{x}}{\frac{3x+5}{x}}$

$y = \frac{2- \frac{7}{x}}{3+ \frac{5}{x}}$

As approaches positive or negative infinity, $\frac{7}{x}$ and $\frac{5}{x}$ both approach 0. Therefore, approaches $\frac{2 }{3 }$ , making the horizontal asymptote the line of the equation $y = \frac{2 }{3 }$ .

Compare your answer with the correct one above

Question

Electrical power can be generated by wind, and the magnitude of power will depend on the wind speed. A wind speed of (in $\tiny m/s$ ) will generate a power of $v^{2}+4v$ . What is the minimum wind speed needed in order to power a device that requires ?

Answer

The simplest way to solve this problem is to plug all of the answer choices into the provided equation, and see which one results in a power of .

Alternatively, one could set up the equation,

$v{^2}+4v-96=0$ and factor, use the quadratic equation, or graph this on a calculator to find the root.

If we were to factor we would look for factors of c that when added together give us the value in b when we are in the form,

.

In our case . So we need factors of that when added together give us .

Thus the following factoring would solve this problem.

Then set each binomial equal to zero and solve for v.

$v+12=0 \rightarrow v=-12$

$v-8=0 \rightarrow v=8$

Since we can't have a negative power our answer is .

Compare your answer with the correct one above

Question

Compared to the graph $y=x^{2}-5$ , the graph $y=(x+4)^{2}-5$ has been shifted:

Answer

The inside the argument has the effect of shifting the graph units to the left. This can be easily seen by graphing both the original and modified functions on a graphing calculator.

Compare your answer with the correct one above

Question

If you look at and $y_{1}=(x-10)^2+10$ on the same graph. What is the transformation that took place from $y\rightarrow y_{1}$ ?

Answer

The graph shifts ten units to the right. The number inside the parentheses shows where and how many units the graph will shift. If it is in the parentheses with the x-coordinate, then it will shift either to the left or right. If the number is negative, it goes to the right. If it is positive, it will shift to the left.

If the number was with the y-coordinate then the graph would shift up and down.

Compare your answer with the correct one above

Question

A triangle is made up of the following points:

$\left { (2,0), (-3,5), (0,4) \right }$

What are the points of the inverse triangle?

Answer

The inverse of a function has all the same points as the original function, except the x values and y values are reversed. The same rule applies to polygons such as triangles.

Compare your answer with the correct one above

Question

Give the -intercept of the graph of the equation $y = \frac{4}{2x+ 5}$ .

Answer

Set in the equation:

$y = \frac{4}{2x+ 5}$

$y = \frac{4}{2 \cdot 0+ 5} = \frac{4}{ 5}$

The -intercept is $\left (0, \frac{4}{5} \right )$ .

Compare your answer with the correct one above

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