Graph Analysis Lab

The slope between two points on a curve is the average rate of change over that interval. As the two points move closer together, the slope approaches the instantaneous derivative.

• v-t graph slope is acceleration.
• x-t graph slope is velocity.
• F-x graph slope is spring constant k for a linear spring.
• p-V graph slope is the inverse compressibility (not always useful).

On the AP exam, when a question asks for the meaning of the slope, answer with the physical quantity and its units, then justify the choice with a derivative argument.

The area between the curve and the x-axis is the accumulation of the y-quantity over the x-range. The units of the area are (y-units) times (x-units).

• v-t area is displacement.
• a-t area is the change in velocity.
• F-x area is work done by the force.
• P-V area is work done by the gas.
• I-t area is total charge transferred.

Always check that the area's units make physical sense. If they do not, you have the wrong axes or the wrong interpretation.

The y-intercept is the value of the dependent variable at x = 0. In physics it almost always represents an initial or zero-state condition.

• v-t y-intercept is initial velocity v₀.
• x-t y-intercept is initial position x₀.
• F-x y-intercept is the force at zero stretch (often zero).
• I-t y-intercept for a discharging capacitor is the initial current I₀.

The x-intercept is where the y-quantity equals zero. For a thrown projectile this is the time of flight back to the ground.

AP physics free-response questions frequently include a graph-reading sub-part. The rubric typically awards:

• One point for stating the correct physical quantity (acceleration, displacement, work, charge).
• One point for correct units.
• One point for a numeric value derived from the graph features (when asked).
• One point for a written justification tied to the derivative or integral meaning.

Always include units. A bare number with no unit is usually worth zero on AP rubrics.