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The vertex form of a quadratic equation is a special way of writing the quadratic function. The general form is given by: \[y = a(x - h)^2 + k\]where

• \(a\) is the coefficient that determines the direction and width of the parabola
• \(h\) represents the x-coordinate of the vertex
• \(k\) represents the y-coordinate of the vertex

To convert a standard quadratic function to vertex form, you need to complete the square. In our problem, we start with: \[f(x) = -x^2 - 8x + 5\]First, factor out the -1 from \(x^2 + 8x\) and rewrite the function as: \[f(x) = - (x^2 + 8x) + 5\]Next, complete the square for the expression inside the parentheses by adding and subtracting the square of half the coefficient of x, which is 16 in this case:\[f(x) = - (x^2 + 8x + 16 - 16) + 5\]This further simplifies to: \[f(x) = - (x + 4)^2 + 16 + 5\]So, the vertex form of the function is: \[f(x) = - (x + 4)^2 + 21\]

The axis of symmetry of a quadratic function is a vertical line that divides the parabola into two mirror images. Its equation is found directly from the vertex form of the quadratic function, specifically from the \(h\) value in \(y = a(x - h)^2 + k\). For our quadratic function: \[f(x) = - (x + 4)^2 + 21\]We know that the vertex form gives us \(h = -4\), hence the axis of symmetry is:\[x = -4\]This line is crucial because it helps in graphing the parabola accurately by providing a reference around which the parabola is symmetrical.

Quadratic functions can either have a maximum or a minimum value, depending on whether the parabola opens upwards or downwards. This behavior is indicated by the coefficient \(a\) in the vertex form \(y = a(x - h)^2 + k\).- If \(a > 0\), the parabola opens upwards, and the function has a minimum value at the vertex.- If \(a < 0\), the parabola opens downwards, and the function has a maximum value at the vertex.For \(f(x) = - (x + 4)^2 + 21\), the coefficient \(a = -1\), which is negative. Therefore, the parabola opens downwards, and the function has a maximum value at the vertex.The maximum value of the function is the y-coordinate of the vertex, which is 21. So, the maximum value of \(f(x)\) is 21.

Graphing a quadratic function involves plotting the vertex, axis of symmetry, y-intercept, and several more key points to get an accurate curve. For \(f(x) = -x^2 - 8x + 5\), follow these steps:1. **Identify the vertex**: The vertex is at \((-4, 21)\), from the vertex form2. **Find the y-intercept**: Set \(x = 0\) in the original function: \[ f(0) = -0^2 - 8(0) + 5 = 5\]So, the y-intercept is (0, 5)3. **Plot additional points**: Calculate more points around the vertex. Some examples are:

• \(x = -3\): \(f(-3) = -(-3)^2 - 8(-3) + 5 = 16\)
• \(x = -5\): \(f(-5) = -(-5)^2 - 8(-5) + 5 = 16\)

4. **Draw the axis of symmetry**: This is the line \(x = -4\), which helps to ensure the parabola is symmetric.5. **Sketch the parabola**: Use the points plotted and the axis of symmetry to draw a smooth, U-shaped curve that opens downwards.By following these steps, you'll have a clear, accurate graph of the quadratic function.