5.9.4 Deciding on Graphing Window

Algebra 15.9.4 Deciding on Graphing Window

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Activity

The equation $m = 20 \cdot (0.8)^{h}$ models the amount of medicine $m$ (in milligrams) in a patient’s body as a function of hours, $h$ , after injection.

1. Without using a graphing tool, decide if the following horizontal and vertical boundaries are suitable for graphing this function.

Be prepared to show your reasoning. $- 10 < h < 100$ $- 100 < m < 1, 000$

Answer: Compare your answer:

No, these boundaries are not suitable for graphing this function.

Horizontal axis: There is probably no medicine left in the body 100 hours after injection, so it is not helpful to graph all the way to an $h$ value of 100. And if there was any medicine in the body before injection, it wouldn’t be accounted for by this equation, so $h$ should probably start at 0.
Vertical axis: The medicine is decreasing in amount, starting from 20 milligrams, so 1,000 for the upper end of the axis is far too large. The negative values for 𝑚 are also not needed because a negative amount of medicine would not make any sense here.

2. Verify your answer by graphing the equation using graphing technology and using the given graphing window. What do you see? Sketch or describe the graph.

Compare your answer:

A graph with axes labeled m (vertical) and h (horizontal) shows black dots rapidly decreasing from the top left and approaching zero as h increases to 100, forming a curve that flattens out. There is a point at (0, 1000), but the dots along the axis go beyond the axis and are tightly together.

3. If your graph in the previous question is unhelpful, modify the window settings so that the graph is more useful. Record the window settings here. Convince a partner why the horizontal and vertical boundaries that you set are better.

Compare your answer:

A reasonable graph would have an $h$ -axis that shows values from 0 to around 20 and an $m$ -axis showing values from 0 to somewhere in the low 20s. Here is an example:

Points plotted on the coordinate plane, showing exponential decay. Horizontal axis from 0 to 22.5, labeled $h$ . Vertical axis from 0 to 22.5, labeled $m$ .

A scatter plot with horizontal axis labeled n (0–20) and vertical axis labeled m (0–20), showing black dots that decrease steeply and gradually level off, forming a downward curve.

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- Authors: Lynn Marecek, MaryAnne Anthony-Smith, Andrea Honeycutt Mathis, Jay Abramson, Sharon North, Amy Baldwin, Alyssa Howell
- Publisher/website: OpenStax
- Book title: Algebra 1
- Publication date: Jun 4, 2025
- Location: Houston, Texas
- Book URL: https://openstax.org/books/algebra-1/pages/about-this-course
- Section URL: https://openstax.org/books/algebra-1/pages/5-9-4-deciding-on-graphing-window

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