Questions: The figure below shows y=3 sin(5x) and y=3 sin(10x). Which graph is y=3 sin(5x) ? Choose one △ Identify the values labeled a to e in the figure. NOTE: Enter exact values or round to three decimal places. a= □ d= □ b= □ e= □ c= □

The figure below shows y=3 sin(5x) and y=3 sin(10x). Which graph is y=3 sin(5x) ? Choose one △ Identify the values labeled a to e in the figure. NOTE: Enter exact values or round to three decimal places. a= □ d= □ b= □ e= □ c= □
Transcript text: The figure below shows $y=3 \sin (5 x)$ and $y=3 \sin (10 x)$. Which graph is $y=3 \sin (5 x)$ ? Choose one $\nabla$ Identify the values labeled $a$ to $e$ in the figure. NOTE: Enter exact values or round to three decimal places. $a=$ $\square$ $d=$ $\square$ $b=$ $\square$ $e=$ $\square$ $c=$ $\square$
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Solution

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Solution Steps

Step 1: Identify the graph of y = 3sin(5x)

The graph of y = 3sin(5x) has a period of 2π/5. The graph of y = 3sin(10x) has a period of 2π/10 = π/5. The solid line completes one full cycle in a larger interval than the dashed line, therefore the solid line is f(x) and corresponds to y = 3sin(5x). The dashed line is g(x) and corresponds to y = 3sin(10x).

Step 2: Find the value of a

'a' represents the x-coordinate where the first minimum of g(x) occurs. Since the period of g(x) is π/5, the first minimum occurs at x = (3π/10). Since we must round to three decimal places: a = 3π/10 ≈ 0.942.

Step 3: Find the value of b

'b' represents the x-coordinate where f(x) first intersects the x-axis after x=0. One full cycle of f(x) = 3sin(5x) is completed at x = 2π/5. Since sin(5x) = 0 when 5x = π, 2π, 3π, ... we see that the first intersection with the x-axis after x = 0 occurs when 5x = π so x = π/5. Therefore, b = π/5 ≈ 0.628

Final Answer:

The graph of y = 3sin(5x) is f(x). a = 0.942 b = 0.628

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