Mini Exam 7
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The table shows values of an exponential function $$f(x)=ab^x$$. Which of the following statements is true?⌄
From the table, $$2,4,8,16,32$$Each output is multiplied by $$2$$So, the common ratio is $$b=2$$ Since $$b$$>$$1$$ the function shows exponential growth.Final answer: $$\text{B}$$
The function $$f$$ is given by $$f(x)=x^2-4$$, and the function $$g$$ is given by $$g(x)=\frac{x+2}{x}$$. Which of the following is an expression for $$f(g(x))$$?⌄
Substitute$$g(x)$$ into $$f(x)$$.$$f(g(x))=(g(x))^2-4$$Replace $$g(x)$$ with $$\frac{x+2}{x}$$Then $$f(g(x))=\left(\frac{x+2}{x}\right)^2-4$$Final answer: $$\text{A}$$
The function $$g$$ is given by $$g(x)=5\sin(2x)$$. Which of the following is an equivalent form of $$g(x)$$?⌄
Use the double-angle identity.$$\sin(2x)=2\sin x\cos x$$Multiply by $$5$$.$$g(x)=5\sin(2x)=5(2\sin x\cos x)$$$$g(x)=10\sin x\cos x$$Final answer: $$\text{A}$$
A complex number has rectangular coordinates $$(-2,2\sqrt{3})$$. Which of the following represents the number in polar form?⌄
Find the modulus.$$r=\sqrt{(-2)^2+(2\sqrt{3})^2}$$$$r=\sqrt{4+12}$$$$r=4$$The point is in Quadrant II.Use the reference angle.$$\tan\theta=\frac{2\sqrt{3}}{-2}=-\sqrt{3}$$So,$$\theta=\frac{2\pi}{3}$$The polar form is:$$4\left(\cos\frac{2\pi}{3}+i\sin\frac{2\pi}{3}\right)$$Final answer: $$\text{A}$$
The function $$f$$ is given by $$f(x)=2x^2-3x+1$$. Which of the following represents the graph of $$f$$ after a vertical dilation by a factor of $$3$$?⌄
A vertical dilation by a factor of $$3$$ multiplies all outputs by $$3$$.$$y=3f(x)$$Substitute $$f(x)$$.$$y=3(2x^2-3x+1)$$Distribute.$$y=6x^2-9x+3$$Final answer: $$6x^2-9x+3$$
The functions $$f$$ and $$g$$ are defined for all real numbers such that $$g(x)=f(3(x+2))$$. Which of the following sequences of transformations maps the graph of $$f$$ to the graph of $$g$$ in the same $$xy$$-plane?⌄
Rewrite the expression.$$g(x)=f(3(x+2))$$First apply the factor inside.This gives a horizontal compression by a factor of $$\frac{1}{3}$$.Then the $$+2$$ shifts the graph left $$2$$ units.Final answer: $$\text{C}$$
Let $$f$$ be a rational function. The numerator has a zero at $$x=2$$ but not at $$x=-3$$. The denominator has zeros at both $$x=2$$ and $$x=-3$$. The multiplicities at $$x=2$$ are equal in numerator and denominator. Which of the following is true?⌄
At $$x=2$$, the zero cancels because multiplicities are equal.This creates a hole.At $$x=-3$$, the denominator is zero but the numerator is not.This creates a vertical asymptote.Final answer: $$\text{C}$$
The table shows values of a function $$P(t)$$. Which model best fits the data?⌄
From the table, $$5,10,20,40,80$$Each value is multiplied by $$2$$So, the model is exponential.The function is $$y=5(2)^t$$Final answer: $$5(2)^t$$