Week 9, Session 1: Slope from Graphs, Points, Tables, and Contexts

Week 9, Session 1: Slope from Graphs, Points, Tables, and Contexts

Self-study purpose

Slope measures steepness and rate of change. On GED® questions, slope can appear in graphs, tables, coordinate pairs, equations, and word problems.

What you should be able to do after this session

• Calculate slope as rise over run.
• Find slope from two points.
• Find slope from a table.
• Interpret positive, negative, zero, and undefined slope.

Visual model

Formula and idea bank

• Slope: \(m=\frac{\text{rise}}{\text{run}}\).
• From points: \(m=\frac{y_2-y_1}{x_2-x_1}\).
• Horizontal line: slope \(0\).
• Vertical line: undefined slope.

Teacher explanation

• Slope is a ratio of vertical change to horizontal change.
• Positive slope rises left to right; negative slope falls left to right.
• In context, slope often means a unit rate, such as dollars per hour or miles per gallon.

Worked example

Find the slope through \((1,3)\) and \((5,11)\): \(m=\frac{11-3}{5-1}=\frac{8}{4}=2\).

GED® problem-solving routine

1. Choose two points.
2. Subtract y-values for rise.
3. Subtract x-values in the same order for run.
4. Simplify the fraction and interpret the sign.

Common mistakes to avoid

• Subtracting x-values on top.
• Changing point order between numerator and denominator.
• Calling vertical slope zero.

Mastery check

Before moving to the next session, complete the 10 questions below without notes. Then correct every missed problem by writing: given information → formula → substitution → answer with units.

Advanced Extension and Harder GED® Practice

Deeper idea

Slope measures steepness as a rate of change: \(\frac{\text{change in }y}{\text{change in }x}\). It can be read from a graph, table, two points, or word context. Positive slope rises left to right; negative slope falls left to right.

Worked example: slope from two points

Find the slope through \((-2,5)\) and \((4,-7)\).

1. Change in \(y\): \(-7-5=-12\).
2. Change in \(x\): \(4-(-2)=6\).
3. Slope: \(\frac{-12}{6}=-2\).

Harder GED® challenge

A water tank loses 18 gallons every 3 minutes. Let \(x\) be minutes and \(y\) be gallons. The slope is \(-18/3=-6\), meaning the amount of water decreases by 6 gallons per minute.

Advanced trap to watch

Keep the same point order in the numerator and denominator. If you start with the second point in \(y_2-y_1\), use the second point first in \(x_2-x_1\) too.