https://file.aiquickdraw.com/mathbot/file/1725509980446eefgzpdd.jpg

The given function in the image is:

\[
f(x) = x^2 - 10x + x
\]

This simplifies to:

\[
f(x) = x^2 - 9x
\]

You're asked to create a worksheet with a graph of this function, with \( x \) values ranging from -10 to 10, and then to choose the graph that most closely resembles the graph of the function based on a table of values.

Let's break this down:

### 1. Calculating Points for the Function:
We need to calculate \( f(x) \) for different \( x \)-values from -10 to 10. I'll generate a few values for clarity.

\[
f(x) = x^2 - 9x
\]

We can then calculate values like:
- \( f(-10) \), \( f(-5) \), \( f(0) \), \( f(5) \), and \( f(10) \).

I'll compute these points to give you the table, and then we'll look at the corresponding graph to identify the correct answer. Let me calculate these values now.Here are the calculated values for the function \( f(x) = x^2 - 9x \):

\[
\begin{array}{c|c}
x & f(x) \\
\hline
-10 & 190 \\
-5 & 70 \\
0 & 0 \\
5 & -20 \\
10 & 10 \\
\end{array}
\]

### Next Step: 
Looking at the options for the graphs, the one that corresponds to this pattern will have the following key features:
- A parabolic shape opening upwards (since the quadratic term \( x^2 \) dominates).
- The graph goes through the points, including \((0, 0)\), \((5, -20)\), and possibly turning upwards around \( x = 10 \).

### Choosing the Graph:
From the graphs shown in the image, Graph 3 should be the correct one, as it fits the behavior of the quadratic function with a minimum and values that match the general trend of the function.

Would you like to explore any more details or clarifications?

Here are some follow-up questions you might find useful:
1. What happens if we change the coefficient of \( x^2 \) in the function?
2. How does the vertex form of a quadratic equation help in sketching graphs quickly?
3. How do the roots of the quadratic function relate to the graph?
4. What is the role of the discriminant in determining the graph's shape?
5. How would the graph change if we added a constant term to the function?

**Tip**: The minimum or maximum of a quadratic function occurs at \( x = \frac{-b}{2a} \), which can help quickly identify key points in the graph.

Create a worksheet with a graph of the function f(x) = x^2 - 10x + x. Choose the graph that looks the closest to the one you created from the options provided.

Learn how to graph the quadratic function f(x) = x^2 - 9x. This step-by-step guide includes table values, graphing techniques, and explanations of key points such as the vertex and the behavior of the parabola. Suitable for students in Grades 9-11.

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