Which function best models the data?

Understand the Problem

The question asks which function best models the data on the number of graduating seniors at Canyon Valley High School from 2012 to 2017. This involves analyzing the provided data to find a linear model that fits the yearly graduate count.

Answer

The function that best models the data is $y = 9.17x + 338.24$.

Steps to Solve

Identify the Data Points The data for graduates from 2012 to 2017 is as follows:

2012: 340
2013: 348
2014: 356
2015: 361
2016: 375
2017: 387

Assign Variables Let $x$ be the number of years since 2012. So, the years translate to:

2012: $x = 0$
2013: $x = 1$
2014: $x = 2$
2015: $x = 3$
2016: $x = 4$
2017: $x = 5$

Corresponding graduate counts become:

$y = 340, 348, 356, 361, 375, 387$

Formulate the Linear Equation We can use linear regression to find the slope ($m$) and intercept ($b$) for the equation $y = mx + b$.
Calculate the Slope (m) Using the formula for the slope: $$ m = \frac{n(\sum xy) - (\sum x)(\sum y)}{n(\sum x^2) - (\sum x)^2} $$
Calculate the Intercept (b) Using the formula for the intercept: $$ b = \frac{\sum y - m(\sum x)}{n} $$
Plug in Values After calculating values and plugging them back, we should find the coefficients for our linear function.
Compare with Given Options After forming the linear equation, we match it with the provided options to find the best fit.

The function that best models the data is $y = 9.17x + 338.24$.

More Information

This equation signifies that for each year after 2012 (where $x=0$), the number of graduates increases by approximately 9.17. The intercept ($338.24$) gives an idea of the starting value of graduates.

Tips

Forgetting to accurately convert years to $x$ values.
Misapplication of the slope and intercept formulas.
Not checking which function format aligns with the derived equation.

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