Use a model to estimate values and interpret the model as a relationship between quantities. | Two-variable data: Models and scatterplots | SAT Math

Core Idea

A model (linear, quadratic, or exponential) expresses a relationship between two quantities. Use it to estimate values by substitution, and interpret it by explaining what the equation says about how the quantities are connected.

Understanding

Once you have a model — whether it's $𝑦 = 3 𝑥 + 12$ , $𝑦 = - 0.5 𝑥 2 + 4 𝑥$ , or $𝑦 = 500 (0.85) 𝑥$ — you can do two things with it: estimate values and describe the relationship.

Estimating values means plugging in. If the model is $𝐶 = 25 𝑡 + 150$ , where $𝐶$ is cost in dollars and $𝑡$ is months, then the estimated cost at month 4 is $𝐶 = 25 (4) + 150 = 250$ dollars.

Interpreting the relationship means explaining what the model structure tells you. That same equation says the cost starts at $150 and grows by $25 each month. For an exponential model like $𝑃 = 500 (0.85) 𝑡$ , the relationship is: the quantity starts at 500 and decreases by 15% each time period.

The SAT will sometimes give you a model and ask what a particular value in the equation represents. Always connect the number to the context: what are the units? What does increasing or decreasing by that amount mean for the real situation?

The model is an approximation. Estimated values are predictions, not exact measurements.

Step by Step

Identify the model equation and what each variable represents in context.
To estimate a value: substitute the known quantity into the equation and solve for the unknown.
To interpret the relationship: describe what happens to y as x changes, using the context and correct units.
For growth/decay models, identify the initial value and the growth/decay factor or rate.

Misconceptions

Treating the model's estimate as an exact measurement. Models produce predictions, which may differ from actual observed values.
Forgetting to include units when interpreting.
Misreading exponential decay as exponential growth. If the base is between 0 and 1 (like 0.85), the quantity is decreasing, not increasing.

Question

Worked Example

The equation $𝑉 = 24, 000 (0.82) 𝑡$ models the value $𝑉$ , in dollars, of a car $𝑡$ years after it was purchased. Which of the following is the best interpretation of the number 0.82 in this equation?

Select an answer to see the explanation