Skills

Slopes and Curve Shapes

Slope is rise over run, the change in the vertical value divided by the change in the horizontal value.

Practice 4: Graphing and Visuals

Part 1 · Hook

Why this matters

A bus fare app shows two lines: one climbing steeply, one barely tilting. Both "go up", but one says fares jump fast with distance and the other says they hardly change. The tilt of a line, its slope, is the number that tells them apart, and the shape of every economics curve is a statement about how two variables move.

Part 2 · Before you start

What this builds on

Part 3 · Prerequisite check

Quick check before you start

1. On a graph, the ordered pair (20, 7) means

  1. 20 across on the horizontal axis and 7 up the vertical axis
  2. 20 up the vertical axis and 7 across
  3. a change of 20 followed by a change of 7
  4. the point halfway between 20 and 7
Show the answer

The horizontal value comes first, then the vertical value.

  • Correct: 20 across on the horizontal axis and 7 up the vertical axis:
  • 20 up the vertical axis and 7 across:
  • a change of 20 followed by a change of 7:
  • the point halfway between 20 and 7:

2. Two variables rise together. Their relationship is

  1. direct
  2. inverse
  3. unrelated
  4. reversed
Show the answer

A direct relationship means both rise or fall together.

  • Correct: direct:
  • inverse:
  • unrelated:
  • reversed:

Part 4 · See it

See it first

Four small graphs: an upward-sloping line (positive slope), a downward-sloping line (negative slope), a vertical line and a horizontal line, each with a note on what it means.
Four shapes: upward sloping (positive slope), downward sloping (negative slope), a vertical line and a horizontal line. LevlPrep original diagram.

Part 5 · Step by step

How it works, step by step

  1. You pick two points on a lineyou can find the rise (change in the vertical value) and the run (change in the horizontal value)
  2. You divide the rise by the runyou get the slope, positive if the line rises and negative if it falls
  3. A line is vertical or horizontalone variable stays fixed while the other changes
  4. A curve bows outwardits slope changes along it, getting steeper as it falls

Part 6 · Key ideas

Key ideas

  • Slope = rise over run = change in the vertical value divided by the change in the horizontal value.
  • Upward sloping: positive slope, a direct relationship. Downward sloping: negative slope, an inverse relationship.
  • A vertical line keeps the horizontal value fixed; a horizontal line keeps the vertical value fixed and has a slope of zero.
  • A straight line has one slope everywhere; a bowed curve does not.

Part 7 · Misconception

A common mistake

The wrong idea: Slope is run over rise: how far across for each step up.

What actually happens: Slope is rise over run: the change in the vertical value divided by the change in the horizontal value. Flipping them gives the reciprocal, a different number.

Part 8 · Check yourself

Check yourself

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

1. A straight line passes through the points (2, 8) and (6, 4). What is its slope?

Type a number. A four-function calculator is allowed, as on the exam.

Show the answer

Slope is rise over run: 4−86−2\frac{4 - 8}{6 - 2}=−44= \frac{-4}{4}=−1= -1. The line falls one unit for each unit it moves right.

  • Answer: -1

2. A line on a graph rises from left to right. What is true of its slope?

  1. It is negative
  2. It is positive
  3. It is zero
  4. It is undefined
Show the answer

Rising to the right means the vertical value grows as the horizontal value grows: a positive slope.

  • It is negative: A negative slope falls from left to right.
  • Correct: It is positive: Right: both values rise together.
  • It is zero: A zero slope is a flat line.
  • It is undefined: Rise over run gives a number for any straight line that is not vertical.

3. On a graph with price on the vertical axis and quantity on the horizontal axis, a curve is a vertical line at a quantity of 50. What does it show?

  1. The quantity is 50 at every price
  2. The price is 50 at every quantity
  3. Quantity rises as price rises
  4. Quantity falls as price rises
Show the answer

A vertical line keeps the horizontal value fixed, so the quantity stays at 50 whatever the price.

  • Correct: The quantity is 50 at every price: Right: the horizontal value never changes along a vertical line.
  • The price is 50 at every quantity: That would be a horizontal line at a price of 50.
  • Quantity rises as price rises: That would be an upward-sloping curve.
  • Quantity falls as price rises: That would be a downward-sloping curve.

4. A straight line and a curve that bows outward both run from the vertical axis down to the horizontal axis. How do their slopes differ?

  1. Both have the same slope at every point
  2. The line gets steeper as it moves right; the bowed curve does not
  3. The line's slope is the same at every point; the bowed curve gets steeper as it moves right
  4. The bowed curve has a positive slope and the line a negative slope
Show the answer

A straight line has one slope everywhere. A curve bowed outward starts flat and grows steeper as it falls toward the horizontal axis.

  • Both have the same slope at every point: Only a straight line keeps one slope; the bowed curve changes.
  • The line gets steeper as it moves right; the bowed curve does not: This reverses them.
  • Correct: The line's slope is the same at every point; the bowed curve gets steeper as it moves right: Right: constant for the line, steeper and steeper for the bowed curve.
  • The bowed curve has a positive slope and the line a negative slope: Both fall from left to right, so both slopes are negative.

5. Place point X on the upward-sloping curve at a higher price than the marked starting point.

QuantityPriceDSQP

Work on the graph (it needs JavaScript), then check.

Show the answer

Along an upward-sloping curve, a higher price goes with a larger quantity: X sits up and to the right on S.

  • Key: see the explanation

6. Place point X on the downward-sloping curve at a larger quantity than the marked starting point.

QuantityPriceDSQP

Work on the graph (it needs JavaScript), then check.

Show the answer

Along a downward-sloping curve, a larger quantity goes with a lower price: X sits down and to the right on D.

  • Key: see the explanation

Part 9 · Summary

Summary

Slope is rise over run, the change in the vertical value divided by the change in the horizontal value. A line that rises to the right has a positive slope, one that falls has a negative slope, a horizontal line has a slope of zero, and a vertical line keeps the horizontal value fixed. A straight line has the same slope at every point, while a bowed curve changes slope as you move along it.

Part 10 · Up next

What comes next

Part 11 · Connections

Connections

Question set for this topic