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The vertex of a quadratic function is a crucial point where the function reaches its maximum or minimum value. To find the vertex of a quadratic function of the form \(f(x) = ax^2 + bx + c\), we can use the vertex formula \(x = -\frac{b}{2a}\).
Once we have the x-coordinate, we substitute it back into the quadratic function to find the corresponding y-coordinate.
For the quadratic function given in the exercise, \(f(x) = x^2 + 8x + 20\), follow these steps:

• Identify the coefficients: \(a = 1\), \(b = 8\), and \(c = 20\).
• Use the formula \(x = -\frac{b}{2a}\) to find the x-coordinate: \(x = -\frac{8}{2(1)} = -4\).
• Substitute \x = -4\ back into the function to find the y-coordinate: \(f(-4) = (-4)^2 + 8(-4) + 20 = 4\).

Hence, the vertex of the quadratic function is \(-4, 4)\). This point indicates either a peak (maximum) or a trough (minimum) of the quadratic graph. In this case, because the coefficient of \(x^2\) is positive, the vertex is the minimum point of the function.

The axis of symmetry for a quadratic function is a vertical line that passes through the vertex of the quadratic graph. It splits the parabola into two mirror-image halves.
For any quadratic function expressed as \(f(x) = ax^2 + bx + c\), the equation for the axis of symmetry is given by the x-coordinate of the vertex, which is \(x = -\frac{b}{2a}\).
In our example \(f(x) = x^2 + 8x + 20\), we previously computed the vertex's x-coordinate as \-4\.
Therefore, the axis of symmetry is the line \(x = -4\).

• This line passes through \(-4, y)\) for all values of \(y\).

The axis of symmetry is crucial because it helps us understand the parabolic graph's reflective property, making it easier to plot additional points accurately.

Graphing a quadratic function involves a few simple steps that build on finding the vertex and the axis of symmetry. Here are the steps to graph \(f(x) = x^2 + 8x + 20\):

• Plot the vertex \(-4, 4)\).
• Draw the axis of symmetry \(x = -4\).
• Select additional points on either side of the vertex, calculate their y-values, and plot them on the graph. For example:
  • For \x = -3\, \(f(-3) = 5\), giving the point \(-3, 5)\).
  • For \x = -5\, \(f(-5) = 5\), giving the point \(-5, 5)\).

Plot these points and draw a smooth curve passing through them to form a symmetric parabola.
The parabola will open upwards since the coefficient of \(x^2\) is positive.
By following these steps, you can create a clear and accurate graph of any quadratic function.