Two Variable Data: Models and Scatterplots Pattern - Model Evaluation

Two-Variable Data: Model Evaluation

These questions ask you to choose the best model for a data set — either from a scatterplot (visual) or from a word problem (verbal). The three model types the SAT tests are linear, quadratic, and exponential.

How to Tell the Models Apart

The Key Distinction: Constant Amount vs. Constant Percentage

This is the single most important rule for this pattern:

• Constant amount of change per unit time → Linear. Examples: "adds 50 gallons per hour," "withdraws $75 per week."
• Constant percentage (or factor) of change per unit time → Exponential. Examples: "doubles every year," "decreases by 10% per hour," "grows by a factor of 1.5 each month."

Worked Example 1 — Visual (Exponential Growth)

The scatterplot shows the number of monthly visitors, in thousands, to a new website during its first six months. Which graph shows the most appropriate model?

The four answer choices show: A) a quadratic curve, B) a straight line, C) an exponential growth curve that fits well, D) an exponential curve that rises too steeply.

SOLUTION

The data starts slow (5, 8, 11 thousand) then accelerates sharply (18, 30, 45 thousand). This steepening-over-time pattern is the hallmark of exponential growth — not a straight line, and not a symmetric U-shape.

A) The quadratic curve doesn't capture the continuously steepening nature.
B) A straight line badly underestimates the later months.
D) This exponential curve is too steep — it rises above all the data points in later months.
C) This exponential curve passes through the middle of the data — some points above, some below.
Answer: C

Worked Example 2 — Visual (Exponential Decay)

The scatterplot shows the remaining mass, in grams, of a radioactive substance over 50 years. Which graph shows the most appropriate model?

The four answer choices show: A) a negative linear model, B) an exponential decay curve that fits well, C) an exponential decay curve that is systematically too high, D) a positive linear model.

SOLUTION

The mass decreases over time, but the rate of decrease slows down — the curve flattens out as mass approaches zero. This is classic exponential decay.

A) A negative linear model would decrease at a constant rate — eventually going below zero, which makes no physical sense for mass.
D) A positive linear model goes the wrong direction entirely.
C) This decay curve is systematically above the data points — poor fit.
B) This decay curve passes through the middle of the data with a balanced scatter.
Answer: B

Worked Example 3 — Verbal (Linear)

A water tank contains 200 gallons. A pump adds water at a constant rate of 50 gallons per hour. What type of function best models the amount of water over time?

A) Increasing linear $\quad$ B) Decreasing linear
C) Increasing exponential $\quad$ D) Decreasing exponential

SOLUTION

"Constant rate" → linear. Water is being added → increasing.
Answer: A) Increasing linear

C) would be correct if the tank's volume doubled each hour (constant percentage), but "50 gallons per hour" is a constant amount.

Common Traps

• Choosing exponential when the rate is constant. "50 gallons per hour" is a fixed amount → linear. Exponential would be "increases by 50% per hour."
• Confusing direction. "Withdraws," "loses," and "consumes" mean decreasing. "Adds," "produces," and "gains" mean increasing. Get the direction right, then match the model type.
• On graph questions, choosing a poor-fit curve. Two answer choices might show the same type of curve (e.g., both exponential), but one fits the data much better. Check whether the curve actually passes near the data points, with roughly equal scatter above and below.
• Picking quadratic when there is no direction change. A quadratic model has a turning point (vertex). If the data only curves in one direction (steepening or flattening but never reversing), it's exponential, not quadratic.

What to Do on Test Day

• For word problems: Look for "constant rate/amount" (→ linear) vs. "doubles/halves/percent change" (→ exponential).
• For scatterplots: If the data is roughly straight → linear. If it curves in one direction only (steepening or flattening) → exponential. If it curves up then down (or vice versa) → quadratic.
• When two choices show the same model type: pick the one whose curve passes closest to the data points with balanced scatter.
• Key formulas to remember:
• Linear: $y = mx + b$ (constant slope)
• Exponential: $y = a \cdot b^x$ (constant ratio between consecutive terms)
• Quadratic: $y = ax^2 + bx + c$ (has a vertex/turning point)