Difference between revisions of "005 Sample Final A, Question 18"

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(Created page with "''' Question ''' Graph the following function, <center><math>f(x) = \left(\frac{1}{3}\right)^{x+1} + 1</math></center> <br> Make sure to label any asymptotes, and at least two...")
 
 
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! Final Answers
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! Foundations
 
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|a) False. Nothing in the definition of a geometric sequence requires the common ratio to be always positive. For example, <math>a_n = (-a)^n</math>
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|1) What is the basic graph of <math> f(x) = \left(\frac{1}{3}\right)^{x+1} + 1</math>?
 
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|b) False. Linear systems only have a solution if the lines intersect. So y = x and y = x + 1 will never intersect because they are parallel.
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|2) How is the graph <math>g(x)=x^3+1</math> obtained from <math>f(x)=x^3</math>?
 
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|c) False. <math>y = x^2</math> does not have an inverse.
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|3) How is the graph <math>g(x)=(x+1)^2</math> obtained from <math>f(x)=x^2</math>?
 
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|d) True. <math>cos^2(x) - cos(x) = 0</math> has multiple solutions.
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|Answer:
 
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|e) True.
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|1) The basic graph is <math>y=\left(\frac{1}{3}\right)^x</math>.
 
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|f) False.  
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|2) The graph of <math>g(x)</math> is obtained by shifting the graph of <math>f(x)</math> up 1 unit.
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|3) The graph of <math>g(x)</math> is obtained by shifting the graph of <math>f(x)</math> to the left by 1 unit.
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Solution:
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{| class = "mw-collapsible mw-collapsed" style = "text-align:left;"
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! Step 1:
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|We start with the basic graph of <math>g(x)=\left(\frac{1}{3}\right)^x</math>.
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|To get the graph of <math>f(x)</math> from <math>g(x)</math>, we shift the graph of <math>g(x)</math> up 2 and to the left by 1.
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{|class = "mw-collapsible mw-collapsed" style = "text-align:left;"
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! Step 2:
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|Two ordered pairs are <math>\left(0, \frac{4}{3}\right)</math> &nbsp; and &nbsp; <math>(-1, 1)</math>. There is a horizontal asymptote at <math>y = 1</math>.
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{|class = "mw-collapsible mw-collapsed" style = "text-align:left;"
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! Final Answer:
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|To get the graph of <math>f(x)</math> from <math>\left(\frac{1}{3}\right)^x</math>, we shift the graph of <math>\left(\frac{1}{3}\right)^x</math> up 1 and to the left by 1.
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[[File:5_Sample_Final_18.png]]
 
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[[005 Sample Final A|'''<u>Return to Sample Exam</u>''']]
 
[[005 Sample Final A|'''<u>Return to Sample Exam</u>''']]

Latest revision as of 10:56, 2 June 2015

Question Graph the following function,

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle f(x) = \left(\frac{1}{3}\right)^{x+1} + 1}


Make sure to label any asymptotes, and at least two points on the graph.


Foundations
1) What is the basic graph of Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle f(x) = \left(\frac{1}{3}\right)^{x+1} + 1} ?
2) How is the graph Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle g(x)=x^3+1} obtained from Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle f(x)=x^3} ?
3) How is the graph Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle g(x)=(x+1)^2} obtained from Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle f(x)=x^2} ?
Answer:
1) The basic graph is Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle y=\left(\frac{1}{3}\right)^x} .
2) The graph of Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle g(x)} is obtained by shifting the graph of Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle f(x)} up 1 unit.
3) The graph of Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle g(x)} is obtained by shifting the graph of Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle f(x)} to the left by 1 unit.


Solution:

Step 1:
We start with the basic graph of Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle g(x)=\left(\frac{1}{3}\right)^x} .
To get the graph of Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle f(x)} from Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle g(x)} , we shift the graph of Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle g(x)} up 2 and to the left by 1.
Step 2:
Two ordered pairs are Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \left(0, \frac{4}{3}\right)}   and   Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle (-1, 1)} . There is a horizontal asymptote at Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle y = 1} .
Final Answer:
To get the graph of Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle f(x)} from Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \left(\frac{1}{3}\right)^x} , we shift the graph of Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \left(\frac{1}{3}\right)^x} up 1 and to the left by 1.

5 Sample Final 18.png

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