Two-Variable Data Analysis

A line of best fit (also called a trend line) is a straight line that best represents the data on a scatterplot. While you don't need to calculate it on the SAT, understanding its concept is important.

The line of best fit:

• Shows the general trend of the data
• Minimizes the distance between the line and all data points
• Can be used to make predictions about values not in the original data set

The slope of a line of best fit represents the rate of change between the two variables. It tells you how much the y-value changes for each unit increase in the x-value.

To find the slope from a line of best fit:

1. Pick two points on the line
2. Use the formula: $\mathrm{slope}=\frac{\mathrm{change\; in\; y}}{\mathrm{change\; in\; x}}=\frac{y_2-y_1}{x_2-x_1}$

For example, if a line passes through points (2, 4) and (5, 10):

$\mathrm{slope}=\frac{10-4}{5-2}=\frac{6}{3}=2$

This means that for each unit increase in x, y increases by 2 units.

You can use a line of best fit to estimate values that aren't in your original data set. This is called interpolation (for values within your data range) or extrapolation (for values outside your data range).

To estimate a value:

1. Find the x-value on the x-axis
2. Draw a vertical line up to the line of best fit
3. Read the corresponding y-value

Remember that predictions become less reliable as you move further from your original data range.

In a linear relationship:

• The data points roughly form a straight line
• The rate of change is constant
• The line of best fit is straight

Linear relationships can be:

• Increasing (positive slope)
• Decreasing (negative slope)
• Constant (zero slope)

In an exponential relationship:

• The data points form a curve
• The rate of change is not constant
• The line of best fit is curved

Exponential relationships can be:

• Increasing (growing exponentially)
• Decreasing (decaying exponentially)

A key characteristic of exponential relationships is that the rate of change increases or decreases by a constant factor over equal intervals.

To identify the type of relationship:

• Look at the shape of the data points
• Check if the rate of change is constant
• Consider the context of the problem

Common contexts for different relationships:

• Linear: Distance vs. time at constant speed, cost vs. quantity at fixed price
• Exponential: Population growth, compound interest, radioactive decay

When interpreting data tables with two columns:

• Look for patterns in how the values change
• Check if the changes are constant (linear) or changing by a factor (exponential)
• Consider how the relationship might be represented as a function

For example, in a table showing time and distance:

This shows a linear relationship because the distance increases by 50 miles for each hour.

When interpreting line graphs:

• Identify the variables and their units
• Look for trends and patterns
• Note any significant changes or turning points
• Consider the rate of change between points

The average rate of change between two points is the slope of the line connecting those points. It represents the average change in y per unit change in x over that interval.

To find the average rate of change:

1. Identify two points
2. Use the slope formula: $\mathrm{average\; rate\; of\; change}=\frac{y_2-y_1}{x_2-x_1}$

For example, if a car travels 100 miles in 2 hours, then 200 miles in 4 hours:

$\mathrm{average\; rate\; of\; change}=\frac{200-100}{4-2}=\frac{100}{2}=50$

The car's average speed is 50 miles per hour.