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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 \( f \)?
| \( x \) | \( f(x) \) |
|---|---|
| 0 | 5 |
| 1 | 15 |
| 2 | 45 |
| 3 | 135 |

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:
\[
2x + y = 19 \\
2x + 2y = 26
\]
What is the value of \( 8x + 6y \)?

First, solve the system of equations:
1. Subtract the first equation from the second:
   \[
   (2x + 2y) - (2x + y) = 26 - 19 \implies y = 7
   \]
2. Substitute \( y = 7 \) into the first equation:
   \[
   2x + 7 = 19 \implies 2x = 12 \implies x = 6
   \]

Now find \( 8x + 6y \):
\[
8x + 6y = 8(6) + 6(7) = 48 + 42 = 90
\]

Thus, the value of \( 8x + 6y \) is **90**.

---

### Question 9:
Which linear equation has exactly one solution?
- A) \( 4x + 8 = 2x \)
- B) \( 4x + 8 = 4x \)
- C) \( 4x + 8 = 2(2x + 4) \)
- D) \( 4x + 8 = 2(2x + 5) \)

To check for one solution, we want an equation where simplifying leads to a single value for \( x \). Let's analyze each:
- A) \( 4x + 8 = 2x \):
  \[
  4x - 2x = -8 \implies 2x = -8 \implies x = -4
  \]
  **One solution**.
- B) \( 4x + 8 = 4x \):
  Simplifies to \( 8 = 0 \), which is false, so **no solution**.
- C) \( 4x + 8 = 4x + 8 \):
  Simplifies to \( 8 = 8 \), so **infinite solutions**.
- D) \( 4x + 8 = 4x + 10 \):
  Simplifies to \( 8 = 10 \), which is false, so **no solution**.

Thus, the correct answer is **A)**, as it has exactly one solution.

---

Let me know if you want further details or have any other questions! Here are some related questions:

1. How would you solve a quadratic equation by factoring?
2. What is the general form of an exponential function?
3. Can you describe how to graph absolute value functions?
4. How do you solve systems of equations by substitution?
5. What is the difference between no solution and infinitely many solutions for a system of linear equations?

**Tip:** Always check for domain restrictions when dealing with absolute value and square root functions!

For an exponential function f, the table shows several values of x and their corresponding values of f(x). The given table: | x | f(x) | 0 | 5 | 1 | 15 | 2 | 45 | 3 | 135 |. Which of the following could define f(x)? A) f(x) = 15^x, B) f(x) = 5(3^x), C) f(x) = 5^x, D) f(x) = 3^x.

Learn how to identify an exponential function from a table of values. This problem covers recognizing patterns in exponential growth, suitable for students in Grades 9-11.

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