Windows, Graphs and Key Features

In this lesson you will: Choose a viewing window that contains the part of the graph you need; Read intercepts, maximums and minimums off a graph; Recognize when the window is hiding the answer.

The default window is a guess, not an answer

A graphing calculator opens on some standard window, often roughly -10 to 10 in both directions. There is nothing special about it. It is simply where the machine starts.

If a model says a population reaches 4,000 after 12 years, a window that stops at 10 shows you a blank corner of a much bigger picture.

Set the window from the question

Read the question before you set the window. It usually tells you the range that matters: \"over the first ten years\", \"for prices between $5 and $40\", \"until the ball lands\".

Set the horizontal range to that interval, then set the vertical range wide enough to contain the outputs over it.

What to read off

Three features answer most graph questions: where the curve crosses the vertical axis, where it crosses the horizontal axis, and the highest or lowest point.

The vertical intercept is the starting value. The horizontal intercepts are the zeros. A maximum or minimum is the turning point of a parabola.

Common mistake — Trusting a picture the window has cropped

A question asks for the maximum value of a model. The student graphs it, sees the curve rising to the right edge of the screen, and answers with the value at that edge.

It is a reasonable reading of what is on the screen. The curve really is highest there — within the window.

But the window was never part of the problem. The true maximum may be off to the right, or the curve may already have turned and come back down outside the view.

The same error hides a second solution: a quadratic with two zeros can look like it has one if the window cuts off half the parabola.

How to catch it: Zoom out once before you answer. If the shape changes when you widen the window, you had not seen the whole picture. A curve that runs off the edge of the screen has not been fully examined.