Which function best models the data?

Steps to Solve

1. Identify the Variables

Let $x$ represent the number of years since 2012. Therefore, we can assign:

• For 2012, $x = 0$, Graduates = 340
• For 2013, $x = 1$, Graduates = 348
• For 2014, $x = 2$, Graduates = 356
• For 2015, $x = 3$, Graduates = 361
• For 2016, $x = 4$, Graduates = 375
• For 2017, $x = 5$, Graduates = 387

2. Calculate the Differences

Find the difference in graduates from year to year:

• 348 - 340 = 8
• 356 - 348 = 8
• 361 - 356 = 5
• 375 - 361 = 14
• 387 - 375 = 12

The average change in the number of graduates is not constant, suggesting a linear model may not fit perfectly but we can approximate.

3. Choose the Correct Model

To determine which function best models the data, we can analyze the function options:

• Option 1: $y = 338.24x + 9.17$ suggests a growth in graduates at roughly 338.24 per year.
• Option 2: $y = 9.17x + 338.24$ suggests a lesser growth (9.17) from a base of 338.24.
• Option 3: $y = -338.24 + 9.17x$ suggests a negative slope with x.
• Option 4: $y = -9.17 + 338.24$ suggests a constant without any change.

Since the data indicates an increase, we compare with option 1, as it suggests a higher slope.

4. Verify with One Point

To see if option 1 models well, we test $x = 0$ (2012): $$ y = 338.24(0) + 9.17 = 9.17 $$
This is not close to actual data (340).

Check with $x = 5$ (2017): $$ y = 338.24(5) + 9.17 = 1691.17 $$
Way off.

In fact, the model needs to be refined to align with actual numbers.

Conclusion: A more refined model needs calculation but $y = 9.17x + 338.24$ best accounts for the uptrend over time in vision of lower slope; still inaccurate.