The question in your image involves creating a scatter plot for Consumer Price Index (CPI) data and identifying a suitable model for the data. Here's a breakdown of how to approach each part:

Part a) Create a Scatter Plot and Choose the Correct Graph

You are asked to plot the data with x equal to the number of years after 1945. This means that the x-values will be years like 0 (for 1945), 1 (for 1946), etc. The scatter plot represents the trend in CPI over time. Based on the image, it seems the question provides three possible graphs, and you need to select the correct one.

To determine which graph is correct:

• Look at the general trend of the CPI. If the data shows a general increase over time (as would be expected for CPI due to inflation), the correct plot would display a positive trend.
• The correct graph should show a curve rather than a straight line since inflation data tends to rise exponentially, meaning the CPI grows faster as time goes on.

Part b) What Type of Function Models the Data?

Next, you're asked to determine the best function to model the data. The options are:

• Cubic $y = ax^3 + bx^2 + cx + d$
• Quartic $y = ax^4 + bx^3 + cx^2 + dx + e$
• Linear $y = mx + b$

CPI data, which generally rises exponentially over time, is typically best modeled by a quartic function (or higher-order polynomial) due to its rapid increase as years go by. So, the quartic option would likely be the best fit.

Part c) Find the Equation

To answer this, you would need to use statistical methods such as regression analysis on the provided CPI data to find the specific coefficients for the quartic model.

Let me know if you'd like further assistance on solving these steps in detail or if you need help with the calculations.

Do you have any further questions? Here are some additional thoughts:

1. How would you plot the scatter data manually or using a graphing tool?
2. Why is a quartic model usually preferred for CPI data over time?
3. What are the differences between cubic, quartic, and linear models in terms of shape and behavior?
4. How would you interpret the coefficients in the final equation found by regression?
5. What might cause a deviation from the expected inflation trend in the CPI data?

Tip: When plotting large datasets, always check if a polynomial regression or exponential function fits better than a linear one, as many real-world phenomena like inflation grow non-linearly over time.