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Advanced Geometry : Coordinate Geometry

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Example Questions

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Example Question #3 : How To Graph An Exponential Function

Give the horizontal asymptote of the graph of the function

$g(x) = 9 \cdot 3^{x}- 3$

Possible Answers:

y = 2

The graph has no horizontal asymptote.

y = 3

y = -3

y = -2

Correct answer:

y = -3

Explanation:

We can rewrite this as follows:

$g(x) = 9 \cdot 3^{x}- 3$

$g(x) = 3^{2} \cdot 3^{x}- 3$

$g(x) = 3^{x+2}- 3$

This is a translation of the graph of $f(x) = 3^{x}$ , which has y = 0 as its horizontal asymptote, to the right two units and down three units. Because of the latter translation, the horizontal asymptote is y = -3 .

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Example Question #1 : How To Graph An Exponential Function

If the functions

$f(x) = e^{x}+ 9$

$g(x)= 4e^{x}- 7$

were graphed on the same coordinate axes, what would be the -coordinate of their point of intersection?

Round to the nearest tenth, if applicable.

Possible Answers:

The graphs of and would not intersect.

2.7

14.3

1.7

5.3

Correct answer:

14.3

Explanation:

We can rewrite the statements using for both f(x) and g(x) as follows:

$y = e^{x}+ 9$

$y = 4e^{x}- 7$

To solve this, we can multiply the first equation by , then add:

$-4y = -4e^{x}-36$

$\underline{y = 4e^{x} \; \; - 7}$

-3y =

$-3y \div (-3) = -43 \div (-3 )$

$y \approx 14.3$

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Example Question #5 : How To Graph An Exponential Function

If the functions

$f(x) = e^{x}+ 9$

$g(x) = 4e^{x}- 7$

were graphed on the same coordinate axes, what would be the -coordinate of their point of intersection?

Round to the nearest tenth, if applicable.

Possible Answers:

The graphs of and would not intersect.

1.7

14.3

2.7

5.3

Correct answer:

1.7

Explanation:

We can rewrite the statements using for both f(x) and g(x) as follows:

$y = e^{x}+ 9$

$y = 4e^{x}- 7$

To solve this, we can set the expressions equal, as follows:

$4e^{x}- 7 = e^{x}+ 9$

$4e^{x}- 7 - e^{x} + 7 = e^{x}+ 9 - e^{x} + 7$

$3e^{x} = 16$

$3e^{x}\div 3 = 16 \div 3$

$e^{x} = \approc 5.3333$

$x \approx \ln 5.333 \approx 1.7$

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Example Question #131 : Advanced Geometry

Find the range for,

$y=3^{x}-2.$

Possible Answers:

y< -2

$y\leq -2$

y>-2

$y\geq -2$

Correct answer:

y>-2

Explanation:

$The\ graph\ of\ the\ function\ y=3^{x}-2\ has\ an\ asymptote\ at\ y=-2.$

$An\ asymptote\ on\ a\ graph\ is\ a\ value\ that\ a \ function\ approaches\ but\ never\ touches.$

$Since\ this\ graph\ approaches\ but\ never\ touches\ -2\ we\ use\ the >\ symbol.$

$\\Also\ because\ the\ 3\ is\ positive\ that\ means\ the\ graph\ is\ concave\ up\ so\ >\ is\ chosen\ instead\ of\ <.$

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Example Question #131 : Advanced Geometry

An important part of graphing an exponential function is to find its -intercept and concavity.

Find the -intercept for

$y=2(3)^{x}-5$

and determine if the graph is concave up or concave down.

Possible Answers:

$y-int: (0,-5); Graph\ is\ concave\ up.$

$y-int: (0,-3); Graph\ is\ concave\ up.$

$y-int: (0,-5); Graph\ is\ concave\ down.$

$y-int: (0,-3); Graph\ is\ concave\ down.$

Correct answer:

$y-int: (0,-3); Graph\ is\ concave\ up.$

Explanation:

$\\The\ concavity\ of\ an\ exponential\ graph\ can\ be\ determined\ by\ its\ a-value.$

$This\ function\ is\ in\ the\ form\ y=a(b)^{x}+k.$

$\ So\ the\ a\ value\ is\ 2, based\ on\ the\ function\ we\ were\ given\ y=2(3)^{x}-5$

$If\ the\ a\ value\ is\ positive,\ the\ graph\ will\ be\ concave\ up$ .

$If\ the\ a\ value\ is\ negative,\ the\ graph\ will\ be\ concave\ down.$

$To\ find\ the\ y-intercept,\ plug\ in\ x=0, and\ keep\ in\ mind\ order\ of\ operations,$

$\ Exponent first, then\ multiply.$

$y=2(3)^{0}-5$

y=2(1)-5=-3

$The\ y-intercept\ is\ (0,-3).$

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Example Question #41 : Graphing

Give the equation of the vertical asymptote of the graph of the equation $f(x) = 2e^{x-3}+ 7$ .

Possible Answers:

x= 3

x= 2

x= 7

The graph of f(x) has no vertical asymptote.

x= e

Correct answer:

The graph of f(x) has no vertical asymptote.

Explanation:

Define $g(x) = e^{x}$ . In terms of g(x) , f(x) can be restated as

f(x) = 2g( x-3 )+ 7

The graph of f(x) is a transformation of that of g(x) . As an exponential function, g(x) has a graph that has no vertical asymptote, as g(x) is defined for all real values of ; it follows that being a transformation of this function, f(x) also has a graph with no vertical asymptote as well.

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Example Question #131 : Advanced Geometry

Give the equation of the horizontal asymptote of the graph of the equation $f(x) = 2e^{x-3}- 7$ .

Possible Answers:

y = 2

y = 3

y = -7

y = e

The graph of f(x) has no horizontal asymptote.

Correct answer:

y = -7

Explanation:

Define $g(x) = e ^{x}$ . In terms of g(x) , f(x) can be restated as

$f(x) = 2e^{x-3}- 7$ .

The graph of g(x) has as its horizontal asymptote the line of the equation y = 0 . The graph of f(x) is a transformation of that of g(x) - a right shift of 3 units ( x-3 ), a vertical stretch ( ), and a downward shift of 7 units ( ). The right shift and the vertical stretch do not affect the position of the horizontal asymptote, but the downward shift moves the asymptote to the line of the equation y = -7 . This is the correct response.

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Example Question #41 : Coordinate Geometry

Give the -coordinate of the -intercept of the graph of the function

$f(x) = 2 \cdot 10 ^{x} - 8$

Possible Answers:

10,000

$\log 4$

Correct answer:

Explanation:

The -intercept of the graph of f(x) is the point at which it intersects the -axis. Its -coordinate is 0; its -coordinate is f(0) , which can be found by substituting 0 for in the definition:

$f(x) = 2 \cdot 10 ^{x} - 8$

$f(0) = 2 \cdot 10 ^{0} - 8 = 2 \cdot 1 - 8 = 2 - 8 = -6$

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Example Question #41 : Coordinate Geometry

Give the -coordinate of the -intercept of the graph of the function $f(x) = 2e^{x}+ 7$ .

Possible Answers:

The graph of f(x) has no -intercept.

2e+7

$\ln \frac{7}{2}$

Correct answer:

Explanation:

The -intercept of the graph of f(x) is the point at which it intersects the -axis. Its -coordinate is 0; its -coordinate is f(0) , which can be found by substituting 0 for in the definition:

$f(x) = 2e^{x}+ 7$

$f(0) = 2e^{0}+ 7 = 2 \cdot 1 + 7 = 2+7 = 9$ ,

the correct choice.

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Example Question #41 : Coordinate Geometry

Give the -coordinate of the -intercept of the graph of the function $f(x) = 2e^{x}+ 7$ .

Possible Answers:

$\frac{7}{2}$

$\ln \frac{7}{2}$

The graph of f(x) has no -intercept.

Correct answer:

The graph of f(x) has no -intercept.

Explanation:

The -intercept(s) of the graph of f(x) are the point(s) at which it intersects the -axis. The -coordinate of each is 0,; their -coordinate(s) are those value(s) of for which f(x) = 0 , so set up, and solve for , the equation:

f(x)= 0

$2e^{x}+ 7 = 0$

Subtract 7 from both sides:

$2e^{x}+ 7 - 7 = 0 - 7$

$2e^{x}= - 7$

Divide both sides by 2:

$\frac{2e^{x}}{2}=\frac{ - 7}{2}$

$e^{x} =-\frac{ 7}{2}$

The next step would normally be to take the natural logarithm of both sides in order to eliminate the exponent. However, the negative number $-\frac{ 7}{2}$ does not have a natural logarithm. Therefore, this equation has no solution, and the graph of f(x) has no -intercept.

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Advanced Geometry : Coordinate Geometry

Study concepts, example questions & explanations for Advanced Geometry

All Advanced Geometry Resources

Example Questions

Example Question #3 : How To Graph An Exponential Function

Example Question #1 : How To Graph An Exponential Function

Example Question #5 : How To Graph An Exponential Function

Example Question #131 : Advanced Geometry

Example Question #131 : Advanced Geometry

Example Question #41 : Graphing

Example Question #131 : Advanced Geometry

Example Question #41 : Coordinate Geometry

Example Question #41 : Coordinate Geometry

Example Question #41 : Coordinate Geometry

All Advanced Geometry Resources

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