7.2 Linear Functions, Slope, and Graphs
Key Takeaways
- The GED assesses slope from graphs, equations, and tables, so practice moving among all three representations.
- In y = mx + b, m is the slope or rate of change and b is the y-intercept or starting value.
- A proportional relationship has a graph through the origin and an equation in the form y = kx.
- A relation is a function only when each input has exactly one output.
- Function notation such as f(3) means substitute 3 for x, not multiply f by 3.
GED Focus
The GED assessment targets for graphs and functions include locating points, finding slope from a graph, equation, or table, graphing two-variable linear equations, and evaluating linear or quadratic functions written in function notation. These items are usually not abstract. They describe pay rates, distance over time, membership fees, savings plans, or other rates that can be shown by a line.
Read y = mx + b Like A Sentence
A linear equation in slope-intercept form is y = mx + b. The slope m tells how much y changes when x increases by 1. The y-intercept b tells the value of y when x = 0.
| Representation | What To Look For | Example |
|---|---|---|
| Equation | Coefficient of x | y = 4x + 9 has slope 4 |
| Graph | Rise over run | Up 6, right 3 gives slope 2 |
| Table | Change in y divided by change in x | x rises by 2, y rises by 10 gives slope 5 |
| Verbal model | Unit rate | $12 per hour means slope 12 |
Worked Example: Table To Equation
A tutoring service charges a sign-up fee plus an hourly rate.
| Hours x | Cost y |
|---|---|
| 1 | 45 |
| 2 | 70 |
| 3 | 95 |
The cost rises by 25 each time the hours increase by 1, so the slope is 25. Use y = mx + b with one point, such as (1, 45): 45 = 25(1) + b. Then b = 20. The equation is y = 25x + 20. In context, 20 is the sign-up fee and 25 is the hourly rate.
Functions And The One Output Rule
A function assigns each input exactly one output. In a table, no x-value can appear with two different y-values. On a graph, use the vertical line test: if a vertical line touches the graph more than once, the relation is not a function.
Function notation is just a compact way to name outputs. If f(x) = 2x + 7, then f(5) means replace x with 5: f(5) = 2(5) + 7 = 17. The notation does not mean f times 5.
Worked Example: Compare Slopes
Plan A costs y = 15x. Plan B is shown by a table: at 2 months the cost is 40, and at 5 months the cost is 100. Plan A has slope 15. Plan B has slope (100 - 40) / (5 - 2) = 60 / 3 = 20. Plan B increases faster, even if it might start lower or higher depending on the intercept.
Test-Day Graph Checklist
- Identify what x and y measure before calculating.
- Find the y-intercept from x = 0, not from the first listed point unless x actually equals 0.
- Use rise over run: change in y divided by change in x.
- For proportional relationships, check whether the graph passes through (0, 0).
- For functions, check repeated inputs or use the vertical line test.
When a question asks which representation has the greater rate of change, compare slopes only. Intercepts tell starting values, not speed of increase. If one line starts at 50 and rises by 2 while another starts at 0 and rises by 8, the second has the greater slope even though the first may be higher at small x-values.
GED graph questions reward careful reading. A slope is not just a number; it is a rate with units, such as dollars per hour, miles per minute, or points per assignment.
A line passes through (2, 9) and (6, 21). What is its slope?
Which table represents a function?