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Let's solve these two problems step by step:

### First Problem (Table Transformation):

You are given a function \( f(x) \) in the table, and you are asked to fill in the new table for the function \( g(x) = \frac{1}{2} f(6x) \).

The table provides the following values of \( f(x) \):

| \( x \)   | \( f(x) \) |
|:--------:|:--------:|
| -6       | 18       |
| 0        | 12       |
| 12       | 10       |
| 18       | 8        |

Now, to compute \( g(x) = \frac{1}{2} f(6x) \), we follow these steps:
1. For each value of \( x \), calculate \( 6x \).
2. Use the calculated \( 6x \) values to find \( f(6x) \) from the table.
3. Multiply \( f(6x) \) by \( \frac{1}{2} \) to get \( g(x) \).

Let's go through each \( x \):

- For \( x = -3 \), \( 6(-3) = -18 \). However, the table does not provide \( f(-18) \). So we skip this.
- For \( x = -1 \), \( 6(-1) = -6 \). From the table, \( f(-6) = 18 \), so \( g(-1) = \frac{1}{2} \times 18 = 9 \).
- For \( x = 0 \), \( 6(0) = 0 \). From the table, \( f(0) = 12 \), so \( g(0) = \frac{1}{2} \times 12 = 6 \).
- For \( x = 2 \), \( 6(2) = 12 \). From the table, \( f(12) = 10 \), so \( g(2) = \frac{1}{2} \times 10 = 5 \).
- For \( x = 3 \), \( 6(3) = 18 \). From the table, \( f(18) = 8 \), so \( g(3) = \frac{1}{2} \times 8 = 4 \).

So, the new table for \( g(x) \) is:

| \( x \)   | \( g(x) \) |
|:--------:|:--------:|
| -1       | 9        |
| 0        | 6        |
| 2        | 5        |
| 3        | 4        |

### Second Problem (Graph Transformation):

You are given the graph of the function \( p(t) \) and asked to find the graph for \( h(t) = 5p(t) \).

#### Understanding the transformation:
- The function \( h(t) = 5p(t) \) means that the graph of \( p(t) \) is vertically stretched by a factor of 5. This means every y-coordinate on the graph of \( p(t) \) will be multiplied by 5.
- If the original graph had a slope of 1, it will now have a slope of 5, making it steeper.

Looking at the V-shaped graph, it should retain its V-shape but with a steeper slope on both sides.

Would you like to go into more details or ask any questions?

### Related Questions:
1. How does multiplying a function by a constant affect its graph?
2. What is the effect of compressing a graph horizontally vs vertically?
3. Could we apply a horizontal transformation like \( f(x-2) \) to this function? How would it affect the graph?
4. What happens if we change the transformation to \( h(t) = \frac{1}{2} p(t) \)?
5. How do we approach piecewise functions when applying transformations?

### Tip:
When transforming graphs, visualize vertical and horizontal stretching or compressing to better understand their effects on slopes and intercepts.

Fill in the table considering the equation g(x) = 1/2 * f(6x) and select the graph of h(t) = 5p(t)

Learn how to transform functions by scaling and stretching. Solve the table for g(x) = 1/2 f(6x) and analyze the graph for h(t) = 5p(t). Perfect for high school students covering advanced algebra topics.

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