Q: Given the augmented matrix $$\begin{bmatrix} 1 & 2 & | & 5 \\ 3 & 4 & | & 11 \end{bmatrix}$$, which system of equations does it represent?

$$\begin{cases} x + 2y = 5 \\ 3x + 4y = 11 \end{cases}$$

Question 1

Which of the following is the correct transformation of the exponential function $$f(x) = 2^x to $$obtain $$g(x) = 2$$^{x-3}$$ + 4$$?

Accepted Answer

Shift right by 3 units and up by 4 units

Question 2

A $$\5000$ investment earns 6$$\%$$ annual interest compounded quarterly. What will the investment be worth after 5$ years? Use the formula A = P$$ \left(1 + \frac{r}{n}\$$right)^{nt}$.

Accepted Answer

$$\6$$,728.94$

Question 3

Which of the following is the correct graph for $$f(x) = \log_2$(x-1)$$?

Accepted Answer

Vertical asymptote at $$x = 1$$, domain $$x \u003e 1$$

Question 4

Expand the logarithmic expression $$\log_3$(2x^4$$)$ using properties of logarithms.

Accepted Answer

$$\log_3$(2) + 4\log_3$(x)$$

Question 5

Condense the expression $$2\log(x) + \frac{1}{2}\log(y)$$ into a single logarithm.

Accepted Answer

$$\log(x^2$y^{1/2}$$)$

Question 6

Solve for $$x$$: $$5^{2x-1} = 125$.

Accepted Answer

$$x = 2$$

Question 7

Solve for $$x$$: $$\log_4$(x+2) = 3.$$

Accepted Answer

$$x = 62$$

Question 8

A population of bacteria doubles every $$3$$ hours. If the initial population is $$200$$, what will the population be after $$12$$ hours? Use $$P(t) = P_0$ \cdot 2$$^{t/3}$$.

Accepted Answer

$$3,200$$

Question 9

A radioactive substance decays according to $$A(t) = A_0$ e$$^{-0.2t}$$. If the initial amount is 100$ grams, how much remains after 10$ hours?

Accepted Answer

$$36.79$$ grams

Question 10

Solve the system of equations using any method: $$\begin{cases} 2x + 3y = 7 \ x - y = 1 \end{cases}$$

Accepted Answer

$$x = 2, y = 1$$

Question 11

Given the augmented matrix $$\begin{bmatrix} 1 & 2 & | & 5 \ 3 & 4 & | & 11 \end{bmatrix}$$, which system of equations does it represent?

Accepted Answer

$$\begin{cases} x + 2y = 5 \ 3x + 4y = 11 \end{cases}$$

Question 12

Which of the following is the domain of the function $$f(x) = \ln(x-4)$$?

Accepted Answer

$$x \u003e 4$$

Question 13

What is the equation of the vertical asymptote for the function $$f(x) = \log_{10}(x+5)$$?

Accepted Answer

$$x = -5$$

Question 14

Solve the nonlinear system: $$\begin{cases} x^2$ + y^2$ = 25 \ y = x + 1 \end{cases}$$

Accepted Answer

$$(4, 5)$$ and $$(-3, -2)$$

Question 15

A hospital uses a matrix to track patient medication dosages. If the solution to the matrix equation $$AX = B$$ gives $$X = \begin{bmatrix} 2 \ 3 \end{bmatrix}$$, what does this mean?

Accepted Answer

The first patient receives 2 units, the second patient receives 3 units

Question 16

Which logarithmic property allows you to write $$\log_b(MN) as$$ $$\log_b(M) + \log_b(N)$$?

Accepted Answer

Product property

Question 17

If $$f(x) = 3^x$$ and $$g(x) = 3$$^{x+2}$$, how is the graph of g(x$$)$$ related to f(x$$)$$?

Accepted Answer

Shifted left by 2 units

Question 18

A car's value depreciates exponentially according to $$V(t) = 20,000 \cdot (0.85)^t$$, where $$t is $$years. What is the value after $$4$$ years?

Accepted Answer

$$10,443.13$$

Question 19

Using the DESMOS matrix calculator, which operation would you use to solve the system $$AX = B$$ for $$X$$?

Accepted Answer

Matrix inverse: $$X = A$$^{-1}$$B$$

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