Skills Beta

Slopes and Areas of Graphs

Label every axis with a quantity and unit.

Practice 1: Creating RepresentationsPractice 2: Mathematical Routines

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Question set for this topic

Part 1 · Hook

Why this matters

A graph is a picture of how one quantity depends on another, and two features of it carry most of the physics: how steep it is and how much area sits under it. Read those two and you can get a speed, a distance or an acceleration without a single equation.

Part 2 · Before you start

What this builds on

Part 3 · Prerequisite check

Quick check before you start

1. A walker covers 12 m in 6.0 s at a steady speed. What is her speed?

  1. 2.0 m/s
  2. 72 m/s
  3. 0.50 m/s
  4. 18 m/s
Show the answer

Speed = distance ÷ time = 12 m ÷ 6.0 s = 2.0 m/s.

  • Correct: 2.0 m/s:
  • 72 m/s:
  • 0.50 m/s:
  • 18 m/s:

2. If y ∝ x², what happens to y when x triples?

  1. It becomes 9 times as large
  2. It triples
  3. It becomes 6 times as large
  4. It becomes one ninth as large
Show the answer

The factor squared: 3² = 9.

  • Correct: It becomes 9 times as large:
  • It triples:
  • It becomes 6 times as large:
  • It becomes one ninth as large:

Part 4 · See it

See it first

Two graphs. Left: distance walked in meters against time in seconds, a straight line through the origin to 20 m at 10 s. A dashed triangle between the points (2 s, 4 m) and (8 s, 16 m) shows a run of 6 s and a rise of 12 m, so the slope is 12 m divided by 6 s = 2 m/s; the units of a slope are the y unit over the x unit. Right: speed in m/s against time, a horizontal line at 3 m/s. The rectangle under it from 2 s to 8 s is shaded: its area is 3 m/s times 6 s = 18 m, the distance covered; the units of an area are the y unit times the x unit.
The slope is rise over run, in (y unit)/(x unit). The area under a line is height × width, in (y unit) × (x unit). LevlPrep original diagram.

Part 5 · Step by step

How it works, step by step

  1. A graph has a quantity with a unit on each axisits slope and its area each have a unit, built from those two units
  2. You divide the rise by the run between two points on a lineyou get the slope, the rate at which y changes per unit of x
  3. You multiply the height by the width of the region under a graphyou get an area whose unit is (y unit) × (x unit), often a total such as a distance
  4. Data scatter and the relationship may be curvedyou draw a best-fit line, after choosing what to plot so the graph is straight

Part 6 · Key ideas

Key ideas

  • Every axis gets a label and a unit: "Time (s)". Without both, the graph cannot be read.
  • Slope = rise/run = Δy/Δx, from two points on the line (not data points). Its unit is (y unit)/(x unit), and it is a rate.
  • Area under a graph = height × width for a rectangle, ½ × base × height for a triangle. Its unit is (y unit) × (x unit).
  • A best-fit line passes between scattered points, never dot to dot. If the relationship is curved, linearize: for d = ½at², plot d against t², and the slope is ½a.

Part 7 · Misconception

A common mistake

The wrong idea: To find a slope, divide the y-value of a point by its x-value, for example 9.0 m ÷ 10 s for a line through (0 s, 2.0 m) and (10 s, 9.0 m).

What actually happens: A slope uses changes between two points on the line: (9.0 m − 2.0 m) ÷ (10 s − 0 s) = 0.70 m/s. Dividing one point's values works only when the line passes through the origin.

Part 8 · Check yourself

Check yourself

Exam-style questions. Anything you miss goes into your review queue.

Graph

A rising weather balloon

A student tracks a small weather balloon just after release and records its height above the ground every 2 s. The line through the data is the best-fit line, height = 0.50 t + 1.2 (height in meters, t in seconds).

012345670246810Time (s)Height (m)

Measured heightBest-fit line

Data table
Time (s)Measured heightBest-fit line
01.11.2
22.4—
43.3—
64—
85—
106.46.2

1. Using the best-fit line, what is the slope of the graph, with its unit?

Type a number and its unit.

Show the answer

Pick two points on the best-fit line, far apart, not data points: (0 s, 1.2 m) and (10 s, 6.2 m). Slope = rise/run = (6.2 m − 1.2 m) ÷ (10 s − 0 s) = 0.50 m/s.

  • Answer: 0.50 m/s

2. What does the slope of this graph tell you about the balloon?

  1. The rate its height increases (m/s)
  2. How high it is at each moment of its flight
  3. How many seconds it takes to reach 6.2 m
  4. How high above the ground it started
Show the answer

A slope means "change in y per change in x". Here that is meters of height per second: the rate at which the balloon rises.

  • Correct: The rate its height increases (m/s): Right: the slope is a change in height per change in time, so it is the balloon's upward speed.
  • How high it is at each moment of its flight: Height at a moment is read straight off the graph, not from the slope.
  • How many seconds it takes to reach 6.2 m: That is a time read from the horizontal axis, not a rate.
  • How high above the ground it started: The starting height is the vertical intercept, 1.2 m.

3. What does the vertical intercept of the best-fit line, 1.2 m, represent?

  1. How fast the balloon rises
  2. The total height the balloon gained in 10 s
  3. The error in the height measurements made
  4. Its height when timing began
Show the answer

For a line y = mx + b, b is the value of y at x = 0. Timing started with the balloon already 1.2 m up.

  • How fast the balloon rises: That is the slope, 0.50 m/s.
  • The total height the balloon gained in 10 s: The gain is 6.2 m − 1.2 m = 5.0 m, a change, not the intercept.
  • The error in the height measurements made: The intercept is a real starting value: the balloon was released from a height above the ground.
  • Correct: Its height when timing began: Right: the intercept is the value of y when x is zero.

4. Why is the slope found from the best-fit line instead of from the first and last data points?

  1. The data points are not plotted on the graph
  2. Two data points are not enough to give a slope
  3. The line averages the scatter of all six points
  4. The best-fit line passes through each of the data points
Show the answer

Real data scatter. The first and last points here give (6.4 − 1.1) ÷ 10 = 0.53 m/s, while the best-fit line, which balances points above and below it, gives 0.50 m/s from all six measurements.

  • The data points are not plotted on the graph: They are plotted; the issue is that each one carries some random error.
  • Two data points are not enough to give a slope: Any two points give a slope, but two scattered points give a less reliable one than a line fit to all six.
  • Correct: The line averages the scatter of all six points: Right: each point has some measurement error; the best-fit line uses all the data, so its slope is more reliable.
  • The best-fit line passes through each of the data points: A best-fit line usually passes between points, with about as many above as below.

5. A graph has speed (m/s) on the vertical axis and time (s) on the horizontal axis. Which unit does the slope have, and which unit does the area under the line have?

  1. Slope: m; area: m/s²
  2. Slope: m/s; area: m/s
  3. Slope: s/m; area: m·s²
  4. Slope: m/s²; area: m
Show the answer

Slope units = (y unit)/(x unit); area units = (y unit) × (x unit).

  • Slope: m; area: m/s²: These are swapped: a slope divides the units, an area multiplies them.
  • Slope: m/s; area: m/s: The axis unit alone is neither; slope and area combine both axes' units.
  • Slope: s/m; area: m·s²: Slope is y over x, not x over y, and an area multiplies y by x.
  • Correct: Slope: m/s²; area: m: Right: slope = (m/s) ÷ s = m/s²; area = (m/s) × s = m.

6. A cart starts from rest with a steady acceleration, so d = ½at². A student wants a straight-line graph whose slope gives a. What should she plot?

  1. d against t; the slope is the acceleration a
  2. d against t²; the slope is ½a
  3. d against t²; the slope is twice the acceleration, 2a
  4. d² against t; the slope is half the acceleration
Show the answer

Linearize by matching y = mx: y = d, x = t², m = ½a. Then a = 2 × slope.

  • d against t; the slope is the acceleration a: d against t curves upward because d depends on t², so it has no single slope.
  • Correct: d against t²; the slope is ½a: Right: d = (½a)(t²) has the form y = mx with x = t², so the graph is a line through the origin with slope ½a.
  • d against t²; the slope is twice the acceleration, 2a: The slope is the factor multiplying t², which is ½a, so a = 2 × slope, not slope = 2a.
  • d² against t; the slope is half the acceleration: Squaring d does not undo the square on t; this graph is not a line either.

Part 9 · Summary

Summary

Label every axis with a quantity and unit. The slope of a line is Δy/Δx between two points on it, with units (y unit)/(x unit); the area under a graph is height × width, with units (y unit) × (x unit). Fit a line to data, and linearize a curved relationship before fitting.

Part 10 · Up next

What comes next

Part 11 · Connections

Connections