Two-variable data

Digital SAT · Math · Problem-Solving and Data Analysis. The idea in plain words, the rules to remember, the mistake to avoid and a worked example, written and checked by Mega Lecture.

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The idea

Here each data point has two values, x and y, usually shown on a scatterplot. A line of best fit is a straight line drawn through the middle of the points. It does not pass through every point. It gives a predicted y for each x.

Most questions ask what the slope or the y-intercept means in the situation, or ask you to compare an actual value with the predicted value.

Some data fit a curve better than a line. If y changes by the same amount for each step in x, the model is linear. If y is multiplied by the same number for each step in x, the model is exponential.

Rules to remember

Slope = the predicted change in y when x increases by 1 unit
y-intercept = the predicted value of y when x = 0
To predict, substitute the x-value into the equation of the line
A point above the line has an actual y greater than the predicted y; a point below the line has an actual y less than predicted
Linear model y = mx + b: equal differences in y for equal steps in x
Exponential model y = a(b)^x: a is the value when x = 0, and y is multiplied by b for each increase of 1 in x
b = 1 + r for growth at rate r, and b = 1 − r for decay (for example, b = 0.94 means a 6% decrease per step)

Common mistake

Reading the base of an exponential model as the percent change, for example saying 0.85 means an 85% decrease. The percent change is the distance of the base from 1, so 0.85 means a 15% decrease.

Worked example

The value V, in dollars, of a machine t years after it was bought is 400 when t = 0, 300 when t = 1 and 225 when t = 2. Which equation models the data, and by what percent does the value fall each year?

Differences: 300 − 400 = −100 and 225 − 300 = −75. They are not equal, so the model is not linear.
Ratios: 300 ÷ 400 = 0.75 and 225 ÷ 300 = 0.75. They are equal, so the model is exponential with b = 0.75.
The starting value is 400, so V = 400(0.75)^t.
Percent decrease each year = 1 − 0.75 = 0.25 = 25%.

Answer: V = 400(0.75)^t; the value falls by 25% each year.

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