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Question 5:

For an exponential function $f$, the table shows several values of $x$ and their corresponding values of $f(x)$. Which of the following could define $f$?

The answer choices are:

• A) $f(x) = 15^x$
• B) $f(x) = 5(3^x)$
• C) $f(x) = 5^x$
• D) $f(x) = 3^x$

We notice that for each increase in $x$ by 1, $f(x)$ is multiplied by 3. This suggests that $f(x)$ is of the form $5(3^x)$, where 5 is the initial value when $x = 0$ and the factor of 3 applies for each step of $x$. Thus, the correct answer is B.

Question 6:

Solve the equation $2x + |x - 6| = 30$.

To solve:

1. Split the equation into two cases based on the absolute value $|x - 6|$:
   • Case 1: $x \geq 6$, then $|x - 6| = x - 6$: $2x + (x - 6) = 30 \implies 3x - 6 = 30 \implies 3x = 36 \implies x = 12$
   • Case 2: $x < 6$, then $|x - 6| = -(x - 6) = 6 - x$: $2x + (6 - x) = 30 \implies x + 6 = 30 \implies x = 24$ But this solution contradicts $x < 6$, so it is not valid.

Thus, the solution is $x = 12$.

Question 7:

The graph of $y = (x + k)^2$, where $k$ is a constant, is shown. What is the value of $k$?

From the graph, the vertex appears to be at $(-3, 0)$, meaning the equation has been shifted horizontally by 3 units to the left. The equation $y = (x + k)^2$ represents a parabola with a vertex at $(-k, 0)$, so $k = -3$.

Thus, the value of $k$ is -3, and the correct answer is A) -3.

Question 8:

In the system of equations: