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A quadratic function typically forms a U-shaped curve called a parabola when graphed. The shape and direction of this curve are influenced by the coefficient of the \( x^2 \) term in the quadratic equation, which has the general form \( ax^2 + bx + c \). Here’s a brief overview of some essential characteristics of parabolas:

• If the coefficient \( a \) is positive, the parabola opens upwards, resembling a smile.
• If the coefficient \( a \) is negative, like in our example \( f(x) = -x^2 + 6x - 8 \), the parabola opens downwards. This makes it resemble a frown.

Furthermore, the parabola is symmetric, meaning it is identical on either side of a vertical line known as the axis of symmetry. For any quadratic equation in standard form, the axis of symmetry can be found using the formula \( x = -\frac{b}{2a} \). The vertex and the axis of symmetry play crucial roles in defining the parabola’s shape and location on the graph.

The vertex is a special point on a parabola that represents its turning point, which can be a minimum or maximum depending on the parabola's direction. To find the vertex of a quadratic function given in the form \( ax^2 + bx + c \), we use the formula \( x = -\frac{b}{2a} \) to calculate the x-coordinate of the vertex.Once the x-coordinate is known, we can find the corresponding y-coordinate by substituting the x-value back into the original quadratic equation. For instance, in our expression \( f(x) = -x^2 + 6x - 8 \), we find:

• The x-coordinate of the vertex is \( x = -\frac{6}{2(-1)} = 3 \).
• Substitute this back into the function: \( f(3) = -3^2 + 6(3) - 8 = 1 \).
• So, the vertex is \( (3, 1) \).

This vertex is crucial because it tells us the highest point in the parabola when it opens downwards, serving as a guide for sketching the rest of the curve.

Intercepts are points where the graph crosses the x-axis and the y-axis. Finding these intercepts can help give a clear picture of the graph’s location and intersection points. Here's how to identify them in a quadratic function:

• X-Intercepts: These occur where the function's value (\( f(x) \)) is zero. For \( f(x) = -x^2 + 6x - 8 \): \[ -x^2 + 6x - 8 = 0 \]. Use the quadratic formula: \[ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \].
• Solving yields: \( x = 2 \) and \( x = 4 \), giving the x-intercepts \( (2, 0) \) and \( (4, 0) \).
• Y-Intercept: Occurs where \( x = 0 \). Substitute \( x = 0 \) into \( f(x) \): \( f(0) = -8 \), resulting in the y-intercept \( (0, -8) \).

Understanding these intercepts is essential for plotting the function accurately and ensuring the parabola intersects the axes appropriately on a graph.