Key Features from a Graph

In this lesson you will: Find both intercepts of a line and say what each one means in a context; State where a linear function is increasing or decreasing; State where it is positive and where it is negative, and explain why that is a different question.

The two intercepts answer two different questions

The y-intercept is f(0): the output when the input is zero. In a context it is usually the starting amount — the full tank, the opening balance, the height at launch.

The x-intercept is the input that makes the output zero. In a context it is usually the moment something runs out: the tank empties, the debt clears, the object lands.

Direction is not the same as height

Reading left to right, a line with positive slope is increasing everywhere and a line with negative slope is decreasing everywhere. A line has one slope, so it does not change its mind partway along.

Whether f(x) is positive or negative asks something else entirely: is the graph above or below the x-axis at that input? For f(x)=-2x+6 the answer is positive when x<3 and negative when x>3 — while the function decreases the whole time.

Key features in a context

A mower's tank holds 15 gallons and the engine burns 0.75 gallon an hour, so f(t)=15-0.75t gives the gallons left after t hours.

The y-intercept (0,15) is the full tank. The x-intercept (20,0) is the hour the tank runs dry. The function decreases everywhere, because fuel does not come back. And the graph stops at 20: beyond that the formula would report a negative number of gallons, which nothing in the situation can supply.

Common mistake — Calling a decreasing function negative

Asked where f(x)=-2x+6 is negative, a student answers \"everywhere, because the slope is -2.\"

The word negative is doing two different jobs in this chapter, which is the whole source of the confusion. The slope is negative. The outputs are a separate matter.

Check a few: f(0)=6, f(1)=4, f(2)=2. All positive, all on a falling line. The outputs do not turn negative until x>3.

How to catch it: Ask the two questions separately, pointing at the graph as you go. \"Negative\" means below the x-axis at that input. \"Decreasing\" means going downhill as you read left to right. Height, then direction.