Week 3, Session 5: Slope From Graphs and Points

Week 3, Session 5: Slope From Graphs and Points

Self-study purpose

Slope is the language of steepness and rate. This session teaches slope visually and algebraically so students can move between a graph and a formula.

What you should be able to do after this session

• Compute slope from two points.
• Interpret positive, negative, zero, and undefined slopes.
• Use rise over run from a graph.

Visual model

Core concept

Slope measures change in y compared with change in x: \(m=\frac{y_2-y_1}{x_2-x_1}\). Positive slope rises left to right, negative slope falls, zero slope is horizontal, and undefined slope is vertical.

Deep self-study explanation

Slope measures how much y changes for each change in x. It is not just steepness; it is a ratio. A slope of 2 means y increases 2 units for every 1 unit x increases. A slope of -2 means y decreases 2 units for every 1 unit x increases. Horizontal and vertical lines are special: horizontal lines have no y-change, so slope is 0; vertical lines have no x-change, so the slope is undefined.

Step-by-step method

1. Choose two clear points.
2. Find change in y: rise.
3. Find change in x: run.
4. Divide rise by run.
5. Check the sign using the direction of the line.

Guided problem-solving script

Choose two clear points. Write the slope formula before using numbers. Subtract y-values on top and x-values on bottom in the same order. Simplify the fraction. Then use the graph direction as a sign check: rising is positive, falling is negative, horizontal is zero, vertical is undefined.

Worked GED®-style example

Problem: Find the slope through \((1,2)\) and \((4,8)\).

Step 1: Change in y is \(8-2=6\).
Step 2: Change in x is \(4-1=3\).
Step 3: Slope is \(6/3=2\).
Answer: The slope is 2.

Why each step works

The worked example compares output change to input change. The y-values change by 6 while the x-values change by 3, so each x-step is worth 2 y-units. Dividing gives the rate per one x-unit.

Ten guided examples with increasing difficulty

Example 1 - Foundation

Problem: A line rises 9 units and runs 3 units. What is the slope?
Solution:

1. Slope is rise over run.
2. Compute \(9/3=3\).

Answer: The slope is 3.

Example 2 - GED®-level

Problem: Find the slope through \((-2,5)\) and \((4,-1)\).
Solution:

1. Use \(m=(y_2-y_1)/(x_2-x_1)\).
2. Substitute: \(m=(-1-5)/(4-(-2))\).
3. Simplify: \(m=-6/6=-1\).

Answer: The slope is -1.

Example 3 - Challenge

Problem: A line through \((a,3)\) and \((8,15)\) has slope 2. Find \(a\).
Solution:

1. Use the slope formula: \(2=(15-3)/(8-a)\).
2. Simplify the numerator: \(2=12/(8-a)\).
3. Multiply: \(2(8-a)=12\).
4. Solve: \(16-2a=12\), so \(-2a=-4\), and \(a=2\).

Answer: \(a=2\).

Example 4 - Extra foundation

Problem: What is the slope of the line through \((1,2)\) and \((4,8)\)?
Solution:

1. Use \(m=(y_2-y_1)/(x_2-x_1)\).
2. Compute \((8-2)/(4-1)\).
3. Simplify the ratio to 2.
4. Check the trap: Keep the same point order on top and bottom.

Answer: \(2\)

Example 5 - Extra skill builder

Problem: What is the slope of the line through \((-2,5)\) and \((2,-3)\)?
Solution:

1. Use \(m=(y_2-y_1)/(x_2-x_1)\).
2. Compute \((-3-5)/(2-(-2))\).
3. Simplify the ratio to -2.
4. Check the trap: Keep the same point order on top and bottom.

Answer: \(-2\)

Example 6 - Extra GED®-level

Problem: A line rises 6 units and runs 3 units. What is its slope?
Solution:

1. Slope is rise divided by run.
2. Compute \(6/3=2\).
3. The slope is 2.
4. Check the trap: Do not report the rise alone.

Answer: \(2\)

Example 7 - Extra multi-step

Problem: A line falls 4 units and runs 2 units to the right. What is its slope?
Solution:

1. Falling means the rise is negative.
2. Compute \(-4/2=-2\).
3. The slope is -2.
4. Check the trap: A line that falls left to right has negative slope.

Answer: \(-2\)

Example 8 - Extra reasoning

Problem: What is the slope of a horizontal line?
Solution:

1. A horizontal line has no change in y.
2. \(Slope = 0\) divided by run.
3. So the slope is 0.
4. Check the trap: Horizontal means zero slope, not undefined.

Answer: \(0\)

Example 9 - Extra challenge

Problem: What is the slope of a vertical line?
Solution:

1. A vertical line has no change in x.
2. Slope would divide by 0.
3. Division by 0 is undefined.
4. Check the trap: Vertical lines are undefined; horizontal lines are 0.

Answer: undefined

Example 10 - Extra mastery

Problem: Which pair of points has slope 3?
Solution:

1. Compute change in y: \(7-1=6\).
2. Compute change in x: \(2-0=2\).
3. Slope is \(6/2=3\).
4. Check the trap: Check both rise and run.

Answer: \((0,1)\) and \((2,7)\)

Common mistakes to avoid

• Dividing run by rise.
• Changing point order on only one part of the fraction.
• Calling a vertical slope zero.

Mastery routine

Pick two points from a line on graph paper. Count the rise and run, then verify with the slope formula.

Practice ladder for independent study

1. Count rise and run from a graph.
2. Use the slope formula from two points.
3. Classify slopes as positive, negative, zero, or undefined.
4. Write a sentence explaining the slope.
5. Create one wrong slope by reversing the ratio, then correct it.

Correction checklist

• Did I put change in y over change in x?
• Did I keep subtraction order consistent?
• Did I simplify the ratio?
• Did the sign match the graph direction?
• Did I handle horizontal and vertical lines correctly?