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) = 3^{(x+1)} - 2</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+1</math> obtained from <math>f(x)=x</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-3)^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=3^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 right by 3 units.
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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)=3^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> down 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 (-1,-1), and (0,1). There is a horizontal asymptote at <math>y=-2</math>.
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! Final Answer:
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|To get the graph of <math>f(x)</math> from <math>3^x</math>, we shift the graph of <math>3^x</math> down 2 and to the left by 1.
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|Two ordered pairs are (-1,-1), and (0,1). There is a horizontal asymptote at <math>y=-2</math>
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[[File:4_Sample_Final_5.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>''']]

Revision as of 10:50, 2 June 2015

Question Graph the following function,


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


Foundations
1) What is the basic graph of ?
2) How is the graph Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle g(x)=x+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) 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)^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=3^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 right by 3 units.


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)=3^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)} down 2 and to the left by 1.
Step 2:
Two ordered pairs are (-1,-1), and (0,1). 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=-2} .
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 3^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 3^x} down 2 and to the left by 1.
Two ordered pairs are (-1,-1), and (0,1). 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=-2}

4 Sample Final 5.png

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