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The function provided in the image is \( f(x) = -5x^2 - 2x + 3 \), and it seems that a point \( (-5, 3, 5) \) is given, although the meaning of this point is not clear. Let me break down the problem and what might be expected:

### Step-by-Step Explanation:

1. **Function Type**: 
   The function \( f(x) = -5x^2 - 2x + 3 \) is a quadratic function, and its graph will be a parabola. Since the coefficient of \( x^2 \) is negative (\(-5\)), the parabola opens downward.

2. **Key Features of the Quadratic Function**:
   - **Vertex**: The vertex of a quadratic function \( ax^2 + bx + c \) can be found using the formula \( x = -\frac{b}{2a} \). Here, \( a = -5 \) and \( b = -2 \).
   - **Axis of Symmetry**: The x-coordinate of the vertex is also the axis of symmetry for the parabola.
   - **Y-Intercept**: This is the point where \( x = 0 \), which is simply \( f(0) = 3 \).

3. **Finding the Vertex**:
   Let's compute the x-coordinate of the vertex using the formula \( x = -\frac{b}{2a} \):

   \[
   x = -\frac{-2}{2(-5)} = \frac{2}{-10} = -0.2
   \]

   Now, plug this value back into the function to find the y-coordinate of the vertex:

   \[
   f(-0.2) = -5(-0.2)^2 - 2(-0.2) + 3
   \]
   \[
   f(-0.2) = -5(0.04) + 0.4 + 3
   \]
   \[
   f(-0.2) = -0.2 + 0.4 + 3 = 3.2
   \]

   So, the vertex of the parabola is at \( (-0.2, 3.2) \).

4. **Plotting the Graph**:
   - The parabola will open downward, and its highest point (the vertex) is at \( (-0.2, 3.2) \).
   - It crosses the y-axis at \( (0, 3) \).
   - You can also find the x-intercepts by solving \( f(x) = 0 \) using the quadratic formula if needed.

Would you like more details about how to plot this graph, find the x-intercepts, or clarify the meaning of the point \( (-5, 3, 5) \)?

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Here are 5 related questions:
1. How do you calculate the x-intercepts of the quadratic function \( f(x) = -5x^2 - 2x + 3 \)?
2. How does the vertex form of a quadratic equation help in graphing the function?
3. What is the significance of the axis of symmetry in a quadratic function?
4. Can you describe how the leading coefficient affects the shape of the parabola?
5. How can we confirm if the point \( (-5, 3, 5) \) belongs to the parabola?

**Tip**: Remember that the vertex of a parabola can give you critical information about its highest or lowest point, depending on the direction it opens!

What are the key features and graph of the quadratic function f(x) = -5x^2 - 2x + 3?

Explore the key features of the quadratic function f(x) = -5x^2 - 2x + 3. Learn how to find the vertex, axis of symmetry, and plot the graph. This problem is suitable for students in Grades 9-11 and focuses on important algebraic concepts like parabola properties.

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