Cubic Functions: Identifying, Graphing, and Interpreting Worksheet ... - Free Printable

Problem Analysis and Solution

The provided image contains two sections related to cubic functions: Section A involves identifying cubic graphs, and Section B involves labeling a given cubic graph using specific keywords. Let's solve each section step by step.

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#### Section A: Identifying Cubic Graphs

Task: Check all the cubic graphs from the selection below.

A cubic function is of the form \( f(x) = ax^3 + bx^2 + cx + d \), where \( a \neq 0 \). The graph of a cubic function has the following characteristics:
1. It can have up to three real roots (where the graph crosses the x-axis).
2. It can have at most two turning points (local maximum or minimum).
3. As \( x \to \infty \) or \( x \to -\infty \), the graph will either rise or fall depending on the leading coefficient \( a \) and the degree of the polynomial.
4. The graph is smooth and continuous without any sharp corners or breaks.

Now, let's analyze each graph:

1. Top-left graph (red line):
- This is a straight line, which represents a linear function, not a cubic function.
- Not cubic.

2. Top-second graph (blue curve):
- This graph has two turning points and crosses the x-axis three times.
- It matches the characteristics of a cubic function.
- Cubic.

3. Top-third graph (green curve):
- This graph has one turning point and does not cross the x-axis more than once.
- It resembles a quadratic function rather than a cubic function.
- Not cubic.

4. Top-fourth graph (purple curve):
- This graph has two turning points and crosses the x-axis three times.
- It matches the characteristics of a cubic function.
- Cubic.

5. Bottom-first graph (green V-shape):
- This graph is symmetric and has a sharp corner, resembling an absolute value function.
- It is not smooth and does not match the characteristics of a cubic function.
- Not cubic.

6. Bottom-second graph (orange curve):
- This graph has one turning point and does not cross the x-axis more than once.
- It resembles a quadratic function rather than a cubic function.
- Not cubic.

7. Bottom-third graph (red curve):
- This graph has two turning points and crosses the x-axis three times.
- It matches the characteristics of a cubic function.
- Cubic.

8. Bottom-fourth graph (blue U-shape):
- This graph is symmetric and resembles a quadratic function.
- It does not have the necessary turning points or x-intercepts for a cubic function.
- Not cubic.

Final Answer for Section A:
The cubic graphs are:
- Top-second graph (blue curve)
- Top-fourth graph (purple curve)
- Bottom-third graph (red curve)

\[
\boxed{\text{Top-second, Top-fourth, Bottom-third}}
\]

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#### Section B: Labeling the Graph

Task: Label the graph below using the keywords provided: Root, Minimum, Maximum, Y-intercept.

The given graph is a cubic function with the following features:
1. Roots: Points where the graph crosses the x-axis.
2. Minimum: The lowest point on the graph (local minimum).
3. Maximum: The highest point on the graph (local maximum).
4. Y-intercept: The point where the graph crosses the y-axis.

Let's label the graph accordingly:

1. Roots: There are three points where the graph crosses the x-axis. Label these as "Root."
2. Minimum: Identify the lowest point on the graph. Label this as "Minimum."
3. Maximum: Identify the highest point on the graph. Label this as "Maximum."
4. Y-intercept: Identify the point where the graph crosses the y-axis. Label this as "Y-intercept."

Final Answer for Section B:
- The labels should be placed as follows:
- Roots: At the three x-intercepts.
- Minimum: At the lowest point on the graph.
- Maximum: At the highest point on the graph.
- Y-intercept: At the point where the graph crosses the y-axis.

\[
\boxed{\text{Labels placed as described above.}}
\]

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Summary

- Section A: The cubic graphs are the top-second, top-fourth, and bottom-third graphs.
- Section B: Label the graph with "Root," "Minimum," "Maximum," and "Y-intercept" at the appropriate points.