Advanced Geometry : Advanced Geometry

Study concepts, example questions & explanations for Advanced Geometry

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Example Questions

Example Question #131 : Advanced Geometry

Give the horizontal asymptote of the graph of the function 

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

Possible Answers:

\displaystyle y = -2

\displaystyle y = 3

The graph has no horizontal asymptote.

\displaystyle y = 2

\displaystyle y = -3

Correct answer:

\displaystyle y = -3

Explanation:

We can rewrite this as follows:

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

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

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

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

Example Question #132 : Advanced Geometry

If the functions 

\displaystyle f(x) = e^{x}+ 9

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

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

Round to the nearest tenth, if applicable.

Possible Answers:

\displaystyle 2.7

The graphs of \displaystyle f and \displaystyle g would not intersect.

\displaystyle 1.7

\displaystyle 14.3

\displaystyle 5.3

Correct answer:

\displaystyle 14.3

Explanation:

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

\displaystyle y = e^{x}+ 9

\displaystyle y = 4e^{x}- 7

To solve this, we can multiply the first equation by \displaystyle -4, then add:

\displaystyle -4y = -4e^{x}-36

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

\displaystyle -3y =            \displaystyle -43

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

\displaystyle y \approx 14.3

Example Question #133 : Advanced Geometry

If the functions 

\displaystyle f(x) = e^{x}+ 9

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

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

Round to the nearest tenth, if applicable.

Possible Answers:

\displaystyle 1.7

\displaystyle 5.3

\displaystyle 14.3

\displaystyle 2.7

The graphs of \displaystyle f and \displaystyle g would not intersect.

Correct answer:

\displaystyle 1.7

Explanation:

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

\displaystyle y = e^{x}+ 9

\displaystyle y = 4e^{x}- 7

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

\displaystyle 4e^{x}- 7 = e^{x}+ 9

\displaystyle 4e^{x}- 7 - e^{x} + 7 = e^{x}+ 9 - e^{x} + 7

\displaystyle 3e^{x} = 16

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

\displaystyle x \approx \ln 5.333 \approx 1.7

Example Question #12 : How To Graph An Exponential Function

Find the range for,

\displaystyle y=3^{x}-2.

Possible Answers:

\displaystyle y< -2

\displaystyle y\leq -2

\displaystyle y>-2

\displaystyle y\geq -2

Correct answer:

\displaystyle y>-2

Explanation:

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

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

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

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

Example Question #41 : Graphing

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

Find the \displaystyle y-intercept for 

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

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

Possible Answers:

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

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

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

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

Correct answer:

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

Explanation:

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

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

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

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

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

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

\displaystyle \ Exponent first, then\ multiply.

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

\displaystyle y=2(1)-5=-3

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

Example Question #11 : How To Graph An Exponential Function

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

Possible Answers:

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

\displaystyle x= 7

\displaystyle x= 2

\displaystyle x= 3

\displaystyle x= e

Correct answer:

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

Explanation:

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

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

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

Example Question #44 : Graphing

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

Possible Answers:

\displaystyle y = 3

\displaystyle y = -7

\displaystyle y = 2

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

\displaystyle y = e

Correct answer:

\displaystyle y = -7

Explanation:

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

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

The graph of \displaystyle g(x) has as its horizontal asymptote the line of the equation \displaystyle y = 0. The graph of \displaystyle f(x) is a transformation of that of \displaystyle g(x) - a right shift of 3 units ( \displaystyle x-3 ), a vertical stretch ( \displaystyle 2g ), and a downward shift of 7 units ( \displaystyle -7 ). 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 \displaystyle y = -7. This is the correct response.

Example Question #133 : Advanced Geometry

Give the \displaystyle y-coordinate of the \displaystyle y-intercept of the graph of the function 

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

Possible Answers:

\displaystyle 10,000

\displaystyle -6

\displaystyle -8

\displaystyle \log 4

\displaystyle 4

Correct answer:

\displaystyle -6

Explanation:

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

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

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

 

Example Question #134 : Advanced Geometry

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

Possible Answers:

\displaystyle 7

\displaystyle \ln \frac{7}{2}

\displaystyle 2e+7

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

\displaystyle 9

Correct answer:

\displaystyle 9

Explanation:

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

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

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

the correct choice.

Example Question #135 : Advanced Geometry

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

Possible Answers:

\displaystyle 2

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

\displaystyle \ln \frac{7}{2}

\displaystyle \frac{7}{2}

\displaystyle 7

Correct answer:

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

Explanation:

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

\displaystyle f(x)= 0

\displaystyle 2e^{x}+ 7 = 0

Subtract 7 from both sides:

\displaystyle 2e^{x}+ 7 - 7 = 0 - 7

\displaystyle 2e^{x}= - 7

Divide both sides by 2:

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

\displaystyle 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 \displaystyle -\frac{ 7}{2} does not have a natural logarithm. Therefore, this equation has no solution, and the graph of \displaystyle f(x) has no \displaystyle x-intercept.

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