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The vertex of a parabola is a crucial point as it represents the highest or lowest point of the graph, depending on the direction the parabola opens.

For a quadratic function in the form of \(f(x) = ax^2 + bx + c\), the vertex can be found using the vertex formula:

• The x-coordinate of the vertex is given by \(x = -\frac{b}{2a}\).
• To find the y-coordinate, substitute the x-coordinate back into the function: \(f(x) = a(x)^2 + bx + c\).

In our example, the given quadratic function is \(f(x) = -x^{2} + 2x + 5\). Here:

• \(a = -1\)
• \(b = 2\)

Using the vertex formula:
\(x = -\frac{b}{2a} = -\frac{2}{2(-1)} = 1\).
Now, substitute \(x = 1\) back into the function to get the y-coordinate:
\[f(1) = -1(1)^2 + 2(1) + 5 = 6\].
Thus, the vertex is \((1, 6)\).

Remember:

• The vertex is the peak if the parabola opens downwards \((a < 0)\).
• The vertex is the lowest point if the parabola opens upwards \((a > 0)\).

The axis of symmetry is a vertical line that divides the parabola into two mirrored halves. It always passes through the vertex of the parabola.

For any quadratic function in the form \(f(x) = ax^2 + bx + c\), the axis of symmetry is found using the formula:

• \(x = -\frac{b}{2a}\)

In our example, with the function \(f(x) = -x^{2} + 2x + 5\), the coefficients are \(a = -1\), \(b = 2\), and \(c = 5\).
Using the formula, the axis of symmetry is:
\[x = -\frac{2}{2(-1)} = 1\].
Therefore, the axis of symmetry of this parabola is the line \(x = 1\).

Key Points to Remember about the Axis of Symmetry:

• It helps in graphing the quadratic function by ensuring that each point on one side has a mirror point on the other side.
• It simplifies finding additional points for plotting the graph.
• It always passes through the vertex of the parabola.

Graphing a quadratic function involves plotting its parabola on a coordinate plane. Here are the essential steps:

1. **Identify Key Components**: Start by finding the vertex and axis of symmetry using the methods detailed above. For \(f(x) = -x^{2} + 2x + 5\), the vertex is \((1, 6)\) and the axis of symmetry is \(x = 1\).

2. **Plot the Vertex**: Place a point at \( (1, 6) \) on the graph.

3. **Determine Additional Points**: Since the parabola opens downward (because \(a = -1\)), find other points on both sides of the vertex. Choose values like \(x = 0\) and \(x = 2\):

• \(f(0) = -0^2 + 2(0) + 5 = 5\)
• \(f(2) = -2^2 + 2(2) + 5 = 5\)

4. **Draw the Parabola**: Connect the vertex with the additional points smoothly. Ensure the parabola symmetrically mirrors across the axis of symmetry (\(x = 1\)).

More Tips for Successful Graphing:

• If unsure, calculate more points by picking more x-values and finding corresponding y-values.
• Check the direction of the opening: downward if \(a < 0\), upward if \(a > 0\).
• Use the symmetrical property for accuracy.

Understanding how to graph these functions is vital as it visually demonstrates the relationships within the quadratic equation.