Difference between revisions of "004 Sample Final A, Problem 2"

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! Step 1:
 
! Step 1:
 
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|First, we put the equation into standard graphing form. Multiplying the equation <math>y=\frac{1}{3}x^2 + 2x - 3</math> by 3, we get
 
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|<math>3y=x^2+6x-9</math>.
 
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! Step 2:
 
! Step 2:
 
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|Completing the square, we get <math> 3y=(x+3)^2-18</math>. Dividing by 3 and subtracting 6 on both sides, we have
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|<math>y+6=\frac{1}{3}(x+3)^2</math>.
 
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! Step 3:
 
! Step 3:
 
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|From standard graphing form, we see that the vertex is (-3,-6). Also, to find the <math>x</math> intercept, we let <math>y=0</math>. So,
 
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|<math>18=(x+3)^2</math>. Solving, we get <math>x=-3\pm 3\sqrt{2}</math>.  
 
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|Thus, the two <math>x</math> intercepts occur at <math>(-3+3\sqrt{2},0)</math> and <math>(-3-3\sqrt{2},0)</math>.
 
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! Step 4:
 
! Step 4:
 
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|To find the <math>y</math> intercept, we let <math>x=0</math>. So, we get <math>y=-3</math>.
 
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|Thus, the <math>y</math> intercept is (0,-3).
 
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! Final Answer:
 
! Final Answer:
 
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|The vertex is (-3,-6). The equation in standard graphing form is <math>y+6=\frac{1}{3}(x+3)^2</math>.
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|The two <math>x</math> intercepts are <math>(-3+3\sqrt{2},0)</math> and <math>(-3-3\sqrt{2},0)</math>.
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|The <math>y</math> intercept is (0,-3)
 
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[[004 Sample Final A|<u>'''Return to Sample Exam</u>''']]
 
[[004 Sample Final A|<u>'''Return to Sample Exam</u>''']]

Revision as of 12:56, 6 May 2015

a) Find the vertex, standard graphing form, and x-intercepts for
b) Sketch the graph. Provide the y-intercept.

Foundations
Answer:


Solution:

Step 1:
First, we put the equation into standard graphing form. Multiplying the equation 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=\frac{1}{3}x^2 + 2x - 3} by 3, we get
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 3y=x^2+6x-9} .
Step 2:
Completing the square, we get 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 3y=(x+3)^2-18} . Dividing by 3 and subtracting 6 on both sides, we have
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+6=\frac{1}{3}(x+3)^2} .
Step 3:
From standard graphing form, we see that the vertex is (-3,-6). Also, to find the 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 x} intercept, we let 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=0} . So,
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 18=(x+3)^2} . Solving, we get 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 x=-3\pm 3\sqrt{2}} .
Thus, the two 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 x} intercepts occur 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 (-3+3\sqrt{2},0)} 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 (-3-3\sqrt{2},0)} .
Step 4:
To find the 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} intercept, we let 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 x=0} . So, we get 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} .
Thus, the 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} intercept is (0,-3).
Final Answer:
The vertex is (-3,-6). The equation in standard graphing form 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+6=\frac{1}{3}(x+3)^2} .
The two 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 x} intercepts 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 (-3+3\sqrt{2},0)} 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 (-3-3\sqrt{2},0)} .
The 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} intercept is (0,-3)

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