The question states:

1. The population of Crazy Creek, Iowa was 2350 in 2004. In 2012, the population was 2598. Write a linear function to describe the population during this time.

Solution:

To find a linear function that describes the population over time, we will use the slope-intercept form of a linear equation:

$y = mx + b$

where:

• $y$ is the population,
• $x$ is the year,
• $m$ is the slope (rate of change of the population per year), and
• $b$ is the y-intercept (the population at the starting year).

Step 1: Determine the Slope (m)

We are given two data points:

• In 2004: Population = 2350
• In 2012: Population = 2598

First, let's calculate the slope $m$:

$m = \frac{{\text{{Change in population}}}}{{\text{{Change in year}}}} = \frac{{2598 - 2350}}{{2012 - 2004}} = \frac{{248}}{{8}} = 31$

So, the slope $m = 31$.

Step 2: Write the Linear Equation

Now that we have the slope, we can substitute one of the points to solve for $b$. Let's use the point (2004, 2350):

$2350 = 31(2004) + b$

Calculate for $b$:

$2350 = 62224 + b \implies b = 2350 - 62224 = -59874$

So, the y-intercept $b = -59874$.

Step 3: Construct the Linear Function

The linear function that describes the population is:

$y = 31x - 59874$

Conclusion

The linear function describing the population of Crazy Creek, Iowa from 2004 to 2012 is:

$y = 31x - 59874$

Would you like more details or have any questions?

Follow-Up Questions:

1. How would the function change if the population decreased over time?
2. What would be the population in 2015 according to this function?
3. How do you interpret the slope in this context?
4. What does the y-intercept represent in terms of population and time?
5. How would you describe the population change rate if it was non-linear?

Tip:

When working with linear functions, always check your slope and intercept calculations for consistency with the context provided.