Question

for an angle to be in standard position, its vertex is at the origin and the initial side lies on the

a) positive (x)-axis.
b) positive (y)-axis.
c) negative (x)-axis.
d) negative (x)-axis.

question 15 (5 points)
write the equation of a trigonometric function for the sinusoid shown.

a) (y = 2 cos x)
b) (y = 2 sin x + 1)
c) (y = 2 cos x - 1)
d) (y = 2 sin x - 1)

question 16 (5 points)
which point is an (x)-intercept of the function (y = 2 cot(\frac{1}{4}x))?

Explanation:

Identify standard position definition

An angle is in standard position when its vertex is at the origin \((0,0)\) and its initial side lies on the positive \(x\)-axis.

Analyze the sinusoid graph features

• Amplitude: The peak is at \(y = 1\) and the trough is at \(y = -3\). The vertical distance is \(1 - (-3) = 4\), so the amplitude is \(A = \frac{4}{2} = 2\).
• Midline: The vertical shift is \(D = \frac{1 + (-3)}{2} = -1\).
• Y-intercept: At \(x = 0\), the graph is at its maximum value \(y = 1\). A cosine function \(y = A \cos(x) + D\) with \(A = 2\) and \(D = -1\) gives \(y = 2 \cos(0) - 1 = 1\), which matches the graph. Thus, the equation is \(y = 2 \cos x - 1\).

Determine the x-intercept of the cotangent function

• Function: \(y = 2 \cot(\frac{1}{4}x)\)
• Condition for x-intercept: \(y = 0 \implies \cot(\frac{1}{4}x) = 0\)
• Solve for x: \(\cot(\theta) = 0\) when \(\theta = \frac{\pi}{2} + k\pi\) for \(k \in \mathbb{Z}\).

$$ \frac{1}{4}x = \frac{\pi}{2} \implies x = 2\pi $$

Thus, a primary \(x\)-intercept is at \((2\pi, 0)\).

Answer:

Question 14

• A) positive \(x\)-axis. (Correct answer)
• B) positive \(y\)-axis.
• C) negative \(x\)-axis.
• D) negative \(x\)-axis.

Question 15

• A) \(y = 2 \cos x\)
• B) \(y = 2 \sin x + 1\)
• C) \(y = 2 \cos x - 1\) (Correct answer)
• D) \(y = 2 \sin x - 1\)

Question 16

The \(x\)-intercept of the function \(y = 2 \cot(\frac{1}{4}x)\) is <blank>\((2\pi, 0)\)</blank>.