005 Sample Final A, Question 19

Question Consider the following function,

$f(x)=-\sin \left(3x+{\frac {\pi }{2}}\right)+1$

a. What is the amplitude?

b. What is the period?

c. What is the phase shift?

d. What is the vertical shift?

e. Graph one cycle of f(x). Make sure to label five key points.

Foundations:
1) For parts (a) - (d), How do we read the relevant information off 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 A\sin(Bx + C) + D?}
2) What are the five key points when looking 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 \sin(x)?}
Answer:
1) The amplitude is A, the period 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 \frac{2\pi}{B}} , the horizontal shift is left by C units if C is positive and right by C units if C is negative, the vertical shift is up by D if D is positive and down by D units if D is negative.
2) The five key points 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 (0, 0),~ (\frac{\pi}{2}, 1), ~ (\pi, 0), ~ (\frac{3\pi}{2}, 0),~ \text{and } (2\pi, 0).}

Solution:

Step 1:
We can read off the answers for (a) - (d):
Amplitude: -1, period: 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 \frac{2\pi}{3}~} , phase shift: Left by 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 \frac{\pi}{2}~} and vertical shift up by 1.

Step 2:
The five key points for the graph 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 (-\frac{\pi}{6}, 1), (0, 0), (\frac{\pi}{6}, 1), (\frac{\pi}{3}, 2), (\frac{\pi}{2}, 1)}

Step 3:
Now plot the five key points and sketch a graph that looks like 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 \sin} graph through those points.

Final Answer:
The final solution is the portion of the graph that below 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 = \vert x\vert + 1} and inside 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^2 + y^2 = 9}
The region we are referring to is shaded both blue and red.

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