Based on the provided images, we are tasked with finding the exponential equation $y = ab^x$ that best fits the given tuition data over time. Let's proceed step-by-step:

Table Data:

1. Years (x): 2011 to 2018, represented as $x = 0, 1, 2, \dots, 7$.
2. Tuition (y): $26,674, $27,670, $28,124, ..., $35,479.

Process to Find $y = ab^x$:

We will use exponential regression to calculate the values of $a$ (initial value) and $b$ (growth rate). Here's how:

1. Data Points:

   • $(x_0, y_0) = (0, 26674)$
   • $(x_1, y_1) = (1, 27670)$
   • $(x_2, y_2) = (2, 28124)$, and so on, up to $(x_7, y_7) = (7, 35479)$.

2. Exponential Regression: To find $a$ and $b$, you can:

   • Input data into a graphing calculator using STAT > Edit.
   • Perform exponential regression STAT > CALC > ExpReg.

Calculation (Manual or Tool-Based Approximation):

1. Use Regression Software/Tools (e.g., Excel, Python):

   • Fit $y = ab^x$ to the data.
   • Obtain parameters $a$ and $b$.

2. Coefficient of Determination $r^2$:

   • It measures the goodness of fit for the curve to the data.
   • $r^2 = 1$ means perfect fit; closer to $1$, better the fit.

Final Equation:

Once regression is applied, the exponential equation will take the form: $y = a \cdot b^x$ Where:

• $a$ = initial value when $x = 0$,
• $b$ = base that reflects the growth rate.

Would you like me to compute the regression directly using Python, or should I guide you through calculator steps?

Related Questions:

1. What does $a$ and $b$ represent in the context of tuition?
2. How can the exponential model be used to predict future tuition costs?
3. Why is $r^2$ important when fitting a curve?
4. How does exponential growth differ from linear growth in this dataset?
5. Can the same method be used for other datasets with exponential trends?

Tip:

When dealing with exponential data, always check for anomalies that might suggest a better fit with another model, such as linear or logarithmic.