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The axis of symmetry of a quadratic function plays a very important role in understanding its graph. It's a vertical line that divides the parabola into two mirror-image halves.
The formula to find the axis of symmetry is closely linked to the vertex of the parabola. For a quadratic function in the form \( ax^2 + bx + c \), the axis of symmetry is given by the formula \( x = -\frac{b}{2a} \).

Using the provided example: \( f(x) = -x^2 - 6x + 3 \), we identified that \( a = -1 \) and \( b = -6 \). Plugging these into the formula, we get \[ x = -\frac{-6}{2(-1)} = -3 \].

This tells us that the axis of symmetry is \( x = -3 \). Any point along this line will equidistant from corresponding points on the other side of the parabola.

Quadratic functions can have either a maximum or a minimum value, depending on the direction the parabola opens. This is determined by the coefficient of the \( x^2 \) term. If it is positive, the parabola opens upwards, and the vertex is its minimum point. If it's negative, the parabola opens downwards, making the vertex a maximum point.

In our example, the function \( f(x) = -x^2 - 6x + 3 \) has a negative \( x^2 \) coefficient of \( -1 \), indicating that the parabola opens downward. This means the vertex is at a maximum.
We calculate the coordinates of the vertex. The x-coordinate \( h \) is found using \[ h = -\frac{b}{2a} = -3 \].
We substitute \( h = -3 \) back into the function to find the y-coordinate \( k \): \[ k = f(-3) = -(-3)^2 - 6(-3) + 3 = 12 \].
The vertex is at \((-3, 12)\), therefore, the maximum value of the function is 12.

Graphing quadratic functions can be straightforward once you understand a few essential steps.

First, determine the vertex of the parabola. As calculated in our example, the vertex here is \((-3, 12)\). Next, draw the axis of symmetry, which, in this case, is the line \( x = -3 \).

To provide more accuracy, select other points on the parabola on either side of the vertex. One way to do this is by picking x-values and substituting them into the function to find corresponding y-values. For instance, use \( x = -2 \) and \( x = -4 \):

• For \( x = -2 \): \[ f(-2) = -(-2)^2 - 6(-2) + 3 = 7 \]. Thus, point \((-2, 7)\).
• For \( x = -4 \): \[ f(-4) = -(-4)^2 - 6(-4) + 3 = 7 \]. Thus, point \((-4, 7)\).

Plot these points and the vertex on a graph. Using the axis of symmetry as a guide, draw a smooth curve through these points to form the parabola. Since this parabola opens downwards, it should look like a 'U' turned upside down.
These steps illustrate how to transform understanding equations into an accurate visual graph.