Question

A layer-cake recipe contains a note that relates the amount of frosting needed to the number of layers in the cake. The standard cake has 2 layers and requires 2 cups of frosting. To make a cake with more than 2 layers, an additional 1/2 cup of frosting is needed for each additional layer. Graph the function that models the relationship between the number of additional layers in the cake, n, and the number of cups of frosting needed to make the cake, F(n). Select points on the graph to plot them.

Answer 1

To model the relationship between the number of additional layers in the cake, \( n \), and the number of cups of frosting needed, \( F(n) \), we first need to define the function based on the information given in the recipe.

1. **Standard Cake Information**:
- A standard cake with 2 layers requires 2 cups of frosting.

2. **Additional Layers**:
- For each additional layer beyond 2 (i.e., for \( n \) additional layers), an extra 0.5 cups of frosting is required.

With this information, we can express the function \( F(n) \):

- If \( n = 0 \) (no additional layers, i.e., the cake remains at 2 layers), then:
\[
F(0) = 2 \text{ cups of frosting}
\]

- For \( n > 0 \) (one or more additional layers), the total frosting requirement can be modeled as:
\[
F(n) = 2 + 0.5n
\]

### Let's plot some points to visualize the function:

Taking several values of \( n \) (the number of additional layers):

- \( n = 0 \):
\[
F(0) = 2 + 0.5(0) = 2
\]
Point: \( (0, 2) \)

- \( n = 1 \):
\[
F(1) = 2 + 0.5(1) = 2.5
\]
Point: \( (1, 2.5) \)

- \( n = 2 \):
\[
F(2) = 2 + 0.5(2) = 3
\]
Point: \( (2, 3) \)

- \( n = 3 \):
\[
F(3) = 2 + 0.5(3) = 3.5
\]
Point: \( (3, 3.5) \)

- \( n = 4 \):
\[
F(4) = 2 + 0.5(4) = 4
\]
Point: \( (4, 4) \)

- \( n = 5 \):
\[
F(5) = 2 + 0.5(5) = 4.5
\]
Point: \( (5, 4.5) \)

Now we can summarize the points to plot on the graph:
- \( (0, 2) \)
- \( (1, 2.5) \)
- \( (2, 3) \)
- \( (3, 3.5) \)
- \( (4, 4) \)
- \( (5, 4.5) \)

### Graphing the Function:
The graph of \( F(n) = 2 + 0.5n \) will be a straight line starting from the point \( (0, 2) \) with a slope of 0.5. You can plot these points on a coordinate system, where the x-axis represents the number of additional layers \( n \) and the y-axis represents the cups of frosting \( F(n) \).

### Slope:
The slope of the line is 0.5, indicating that for each additional layer of cake \( n \), the frosting requirement increases by 0.5 cups.

This linear function is helpful for calculating the required frosting for any number of layers beyond the standard two.