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 $(-5, 3, 5)$?

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!