Q: The figure shows a circle centered at the origin. The length of arc $$PQ$$ is $$6$$ units and the radius is $$5$$. Which expression gives the distance of point $$Q$$ from the $$y$$-axis?

Arc length formula. $$s=r\theta$$ $$6=5\theta$$ $$\theta=\frac{6}{5}$$ Distance from the $$y$$-axis is the $$x$$-coordinate. $$x=5\cos\theta$$ $$x=5\cos\left(\frac{6}{5}\right)$$ Final answer: $$5\cos\left(\frac{6}{5}\right)$$

Q: The figure shows the graph of the polar function $$r=4\cos(2\theta)$$. If the domain is restricted to $$0\le\theta\le\frac{\pi}{2}$$, which describes the other remaining piece?

For $$0\le\theta\le\frac{\pi}{2}$$, the graph traces two petals. One is in Quadrant I from $$C$$ to $$E$$. The other is in Quadrant II from $$E$$ to $$A$$. Final answer: $$\text{B}$$

Question 1

The function $$f$$ is given by $$f(x)=3\sin(3x)+2\cos(6x)$$. Using the period of $$f$$, how many complete cycles occur on the interval $$0\le x\le600$$?

Accepted Answer

The period of $$\sin(3x)$$ is $$\frac{2\pi}{3}$$ The period of $$\cos(6x)$$ is: $$\frac{2\pi}{6}=\frac{\pi}{3}$$ The least common period is $$\frac{2\pi}{3}$$ Number of cycles: $$\frac{600}{\frac{2\pi}{3}}=\frac{1800}{2\pi}=\frac{900}{\pi}$$ $$\frac{900}{\pi}\approx286$$ Final answer: $$286$$

Question 2

The function $$f$$ is given by $$f(x)=\sin(3x+0.4)$$. The function $$g$$ is defined by $$g(x)=f(x)-0.25$$. What are the zeros of $$g$$ on the interval $$0\le x\le\pi$$?

Accepted Answer

Set $$g(x)=0$$. $$\sin(3x+0.4)=0.25$$ Solve. $$3x+0.4=\sin^{-1}(0.25)$$ and $$3x+0.4=\pi-\sin^{-1}(0.25)$$ Solve for $$x$$. $$x\approx0.256,\;1.021$$ Final answer: $$0.256,\;1.021$$

Question 3

The figure shows a circle centered at the origin. The length of arc $$PQ$$ is $$6$$ units and the radius is $$5$$. Which expression gives the distance of point $$Q$$ from the $$y$$-axis?

Accepted Answer

Arc length formula. $$s=r	heta$$ $$6=5	heta$$ $$	heta=\frac{6}{5}$$ Distance from the $$y$$-axis is the $$x$$-coordinate. $$x=5\cos	heta$$ $$x=5\cos\left(\frac{6}{5}ight)$$ Final answer: $$5\cos\left(\frac{6}{5}ight)$$

Question 4

The figure shows the graph of the polar function $$r=4\cos(2	heta)$$. If the domain is restricted to $$0\le	heta\le\frac{\pi}{2}$$, which describes the other remaining piece?

Accepted Answer

For $$0\le	heta\le\frac{\pi}{2}$$, the graph traces two petals. One is in Quadrant I from $$C$$ to $$E$$. The other is in Quadrant II from $$E$$ to $$A$$. Final answer: $$	ext{B}$$

Question 5

The figure shows the graph of a trigonometric function $$f$$. Which of the following could be an expression for $$f(x)$$?

Accepted Answer

The amplitude is $$3$$ The midline is $$-1$$ The period is $$\pi$$ So, the function has factor $$2$$ inside. The phase shift matches $$\frac{\pi}{4}$$ Final answer: $$3\cos\left(2\left(x-\frac{\pi}{4}\right)\right)-1$$

Question 6

If the function $$f(x)=\sqrt{x-1}$$ and $$g(x)=\frac{1}{x+3}$$ are combined, which of the following would result in a function with domain $$x\ge1,\ x
e-3$$?

Accepted Answer

For $$f(x)+g(x),$$ the domain is the intersection of the two domains.For $$f,$$ $$x\ge1.$$For $$g,$$ $$x
e-3.$$So, the combined domain is: $$x\ge1,$$ since $$x
e-3$$ is already satisfied when $$x\ge1.$$For $$(g\circ f)(x)=g(f(x)),$$ substitute $$f(x)$$ into $$g.$$$$g(f(x))=\frac{1}{\sqrt{x-1}+3}.$$This requires $$x\ge1$$ and $$\sqrt{x-1}+3
e0.$$The second condition is always true, so the domain is $$x\ge1.$$For $$\frac{g(x)}{f(x)},$$ the denominator cannot be zero.So:$$\sqrt{x-1}
e0.$$This means:$$x
e1.$$Thus, the domain becomes $$x$$>$$1,$$ which is different.Therefore, the expressions that have the same domain are $$\mathrm{i\ and\ ii\ only}.$$Final answer: $$\mathrm{i\ and\ ii\ only}$$

Question 7

Consider the function $$g(x)=x^4+2x^3-5x^2-2x+4$$. If $$2+i$$ is a zero, what is the sum of all real zeros?

Accepted Answer

The polynomial has real coefficients. So, if $$2+i$$ is a zero, then $$2-i$$ is also a zero. The sum of all zeros is the opposite of the coefficient of $$x^3$$. $$-(2)=-2$$ The sum of the non-real zeros is $$(2+i)+(2-i)=4$$ So, the sum of the real zeros is $$-2-4=-6$$ Final answer: $$-6$$

Question 8

A function $$g(x)=\frac{x+4}{x-1}$$ is a transformation of $$f(x)=\frac{1}{x}$$. Which description correctly matches the transformation?

Accepted Answer

Rewrite the function. $$g(x)=\frac{x+4}{x-1}$$ Separate the fraction. $$g(x)=1+\frac{5}{x-1}$$ So, start with $$f(x)=\frac{1}{x}$$ Shift right $$1$$ Multiply vertically by $$5$$ Shift up $$1$$ Final answer: $$	ext{Right }1,\ 	ext{vertical stretch by }5,\ 	ext{up }1$$

Mini Exam 9

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