Choose all of the following that are true. This table shows a linear relationship between the the age of a newborn baby in weeks and their weight.

age (weeks)	weight (pounds)
1	                      6
3	                    8
10	                 15
12	                     17

This table shows a linear relationship between the the age of a newborn baby in weeks and their weight.

age (weeks)	weight (pounds)
1	6
3	8
10	15
12	17

Age is a function of weight.

Weight is a function of age.

The weight of the baby increases at a constant rate of 1 lb every week.

The weight of the baby increases at a constant rate of 2 lbs every week.

The weight of the baby increases at a constant rate of ½ lb every week.

Let's analyze the provided data to determine the correct statements.

We have the following pairs of \((\text{age (weeks)}, \text{weight (pounds)})\):

\[
\begin{aligned}
& (1, 6), \\
& (3, 8), \\
& (10, 15), \\
& (12, 17).
\end{aligned}
\]

To check if the relationship is linear, we need to see if there is a constant rate of change (slope) between the pairs. We calculate the rate of change (slope) \( m \) between each pair of points using the formula:

\[
m = \frac{\Delta y}{\Delta x} = \frac{\text{change in weight}}{\text{change in age}}
\]

Let's calculate the slopes between consecutive points:

1. Between \((1, 6)\) and \((3, 8)\):

\[
m = \frac{8 - 6}{3 - 1} = \frac{2}{2} = 1
\]

2. Between \((3, 8)\) and \((10, 15)\):

\[
m = \frac{15 - 8}{10 - 3} = \frac{7}{7} = 1
\]

3. Between \((10, 15)\) and \((12, 17)\):

\[
m = \frac{17 - 15}{12 - 10} = \frac{2}{2} = 1
\]

Since the rate of change is constant (1 lb per week), the relationship is indeed linear.

Now, let's address each statement:

1. **Age is a function of weight.**

   This is not necessarily true because, for a function, each input should map to exactly one output. Here, each weight does not uniquely determine an age since multiple ages can have the same weight.

2. **Weight is a function of age.**

   This is true because each age maps to exactly one weight.

3. **The weight of the baby increases at a constant rate of 1 lb every week.**

   This is true based on our calculations of the slope.

4. **The weight of the baby increases at a constant rate of 2 lbs every week.**

   This is false. The rate is 1 lb per week.

5. **The weight of the baby increases at a constant rate of ½ lb every week.**

   This is false. The rate is 1 lb per week.

Therefore, the correct statements are:

- Weight is a function of age.
- The weight of the baby increases at a constant rate of 1 lb every week.

Do you want details or have any questions? Here are 5 questions you might want to ask:

1. How do you determine if a relationship is linear?
2. Can you explain how to find the rate of change between two points?
3. What does it mean for a function to be one-to-one?
4. How can you graphically represent the relationship between age and weight?
5. Can we use this data to predict the baby's weight at 15 weeks?

**Tip:** When verifying if a relationship is linear, always check if the rate of change (slope) between all pairs of points is constant.

Explore the linear relationship between the age and weight of newborn babies. Understand how to determine if a relationship is linear using the rate of change (slope). This problem involves calculating slopes between data points and analyzing statements about the constant rate of weight increase.

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