Question

Match the rule and classification for each exponential function graph.
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Rule
Classify

Answer 1

I don't have the graphs or the list of rule/classification choices. Please either

- upload the images of the graphs, or
- paste the list of function rules and the images' descriptions (key points, y-intercept, horizontal asymptote, one or two labeled points).

If you want to match them yourself, here are quick steps to identify each rule and classification:

1. Identify the horizontal asymptote (usually y = k). This gives the vertical shift.
2. Find the y-value at one or two x-values (preferably x = 0 and one other integer). If you know the asymptote k, subtract k from the function values to get the form a·b^x (or a·b^(x−h)).
3. Use the value at x = 0 to find a: a = f(0) − k (unless the graph is shifted horizontally).
4. Use a second point (x1, y1) to solve for b: b = (y1 − k)/a)^(1/x1) (if no horizontal shift). If there's a horizontal shift h, you will solve using a·b^(x−h).
5. Classification:
- If base b > 1 → exponential growth.
- If 0 < b < 1 → exponential decay.
- If a < 0 → reflected across horizontal asymptote (watch sign).
- Note vertical stretch/compression from |a|.

Example:
- If asymptote y = 0, f(0) = 2, f(1) = 6 → a = 2, b = 6/2 = 3 → rule y = 2·3^x (growth).
- If asymptote y = 3, f(0) = 5 → a = 2 (since 5 − 3 = 2); if f(1) = 4 → b = (4 − 3)/2 = 1/2 → rule y = 3 + 2·(1/2)^x (decay).

Share the graphs or their key points and I will match each graph to its rule and label it growth/decay.