https://file.aiquickdraw.com/mathbot/file/1726384018078lztfjvrp.jpg

The task in the image is to graph the function \( f(x) = x^3 + 4x^2 - 5 \) by using a table of values.

### Steps:
1. **Create a table of values** by choosing some \( x \)-values and calculating the corresponding \( f(x) \)-values.
   Let's choose \( x = -4, -2, 0, 2, 4 \) based on the grid provided.

2. **Calculate the values of the function**:
   \[
   f(x) = x^3 + 4x^2 - 5
   \]
   For each \( x \), substitute the value into the function:

   - When \( x = -4 \):
     \[
     f(-4) = (-4)^3 + 4(-4)^2 - 5 = -64 + 64 - 5 = -5
     \]
   - When \( x = -2 \):
     \[
     f(-2) = (-2)^3 + 4(-2)^2 - 5 = -8 + 16 - 5 = 3
     \]
   - When \( x = 0 \):
     \[
     f(0) = (0)^3 + 4(0)^2 - 5 = -5
     \]
   - When \( x = 2 \):
     \[
     f(2) = (2)^3 + 4(2)^2 - 5 = 8 + 16 - 5 = 19
     \]
   - When \( x = 4 \):
     \[
     f(4) = (4)^3 + 4(4)^2 - 5 = 64 + 64 - 5 = 123
     \]

3. **Plot the points** on the graph:
   The points you can plot are:
   \[
   (-4, -5), (-2, 3), (0, -5), (2, 19), (4, 123)
   \]
   
4. **Sketch the graph**:
   After plotting these points, you will draw a smooth curve passing through them, noting that this is a cubic function. The shape of a cubic function usually includes a turning point, and it goes to negative infinity as \( x \to -\infty \) and positive infinity as \( x \to +\infty \).

Would you like me to generate the graph for you or explain further details about this? Here are some follow-up questions that can deepen your understanding:

1. How do cubic functions typically behave for large positive and negative values of \( x \)?
2. Why do we include turning points when sketching cubic functions?
3. What are the general characteristics of the graph of a polynomial of degree 3?
4. How would you determine the turning points or local minima/maxima of this cubic function?
5. What is the significance of the leading term \( x^3 \) in determining the graph's overall shape?

**Tip**: For polynomial functions, the degree of the leading term gives important information about the general shape of the graph.

Graph f(x) = x^3 + 4x^2 - 5 by using a table of values.

Learn how to graph the cubic function f(x) = x^3 + 4x^2 - 5 using a table of values. This detailed guide explains key concepts, formulas, and graphing techniques, suitable for high school students in Grades 9-12.

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