7.2 Linear Functions, Slope, and Graphs

GED Focus: Lines Model Real Rates

The GED targets for graphs and functions include locating coordinate points, finding slope from a graph, equation, or table, graphing two-variable linear equations, and evaluating linear or simple quadratic functions written in function notation. These items are concrete: they describe pay rates, distance over time, membership fees, savings plans, or fuel use, all of which a straight line can model.

Graphing items appear in the calculator-allowed section, but the TI-30XS MultiView is a scientific, not graphing, calculator, so it will NOT draw lines for you. You read slope and intercepts yourself.

The coordinate plane has four quadrants. A point such as (-3, 5) sits left of the y-axis and above the x-axis (Quadrant II). Getting the sign of each coordinate right is the foundation for plotting and for reading slope correctly.

Read y = mx + b Like a Sentence

The slope-intercept form is y = mx + b. The slope m is the change in y for each increase of 1 in x. The y-intercept b is the value of y when x = 0. A useful sentence: "start at b, then go up m for every step right."

Slope can be negative (the line falls), zero (a horizontal line, y = b), or undefined (a vertical line, x = a constant).

Worked Example: Table to Equation

A tutoring service charges a one-time sign-up fee plus an hourly rate.

The cost rises by 25 each time x increases by 1, so the slope is 25. Plug one point, say (1, 45), into y = mx + b: 45 = 25(1) + b, which gives b = 20. The equation is y = 25x + 20. In context, the $20 is the sign-up fee (the value at x = 0) and the $25 is the hourly rate. A classic trap is reporting 45 as the fee; 45 is the cost after one hour, not the starting value.

Functions and the One-Output Rule

A function assigns each input exactly one output. In a table, no x-value may pair with two different y-values. On a graph, apply the vertical line test: if any vertical line touches the graph more than once, it is not a function. A circle, for instance, fails the test.

Function notation simply names outputs. If f(x) = 2x + 7, then f(5) means replace every x with 5: f(5) = 2(5) + 7 = 17. The GED also tests reading values from a function table or graph, such as "find f(3)" by locating x = 3 and reading the matching y. It does not mean f multiplied by 5. The GED may also reverse the task and give an output, asking which input produces it: if f(x) = 2x + 7 and the question states f(x) = 19, solve 2x + 7 = 19 to find x = 6.

Worked Example and Test-Day Checklist

Plan A costs y = 15x. Plan B is given 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, regardless of where each plan starts.

Run this checklist on every graph item:

• Identify what x and y measure, with units, before calculating.
• Find the y-intercept at x = 0, not from the first listed point unless x is actually 0 there.
• Compute slope as rise over run: change in y divided by change in x.
• For proportional relationships, confirm the line passes through (0, 0).
• To compare rates of change, compare slopes only; intercepts give starting values, not speed. A line starting at 50 rising by 2 grows slower than a line starting at 0 rising by 8.

Proportional vs. Non-Proportional

A proportional relationship has the form y = kx with no constant added, so its graph passes through the origin (0, 0) and k is both the slope and the unit rate. "$0.12 per text message" gives y = 0.12x, a proportional line through the origin. As soon as there is a flat fee, such as a $20 sign-up cost, the relationship is non-proportional (y = 25x + 20) and the line no longer hits the origin. The GED tests this distinction directly: a question may show a table and ask whether it is proportional, which is true only when every y/x ratio is identical and the line would pass through (0, 0).

Always check the ratio at two or more points before deciding.