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The vertex of a parabola is a critical point that often represents its highest or lowest point, depending on the direction the parabola opens. For the quadratic function \( f(x) = ax^2 + bx + c \), the vertex can be found using the formula \( x = -\frac{b}{2a} \). This formula calculates the x-coordinate of the vertex. Once you have the x-coordinate, substitute it back into the original equation to find the corresponding y-coordinate.

In our example \( f(x) = x^2 - 6x \), where \( a = 1 \), and \( b = -6 \), we compute the vertex as follows:

\[ x = -\frac{-6}{2 \times 1} = 3 \]

Substituting \( x = 3 \) back into the equation:

• \( f(3) = 3^2 - 6 \times 3 = -9 \)
• So, the vertex is \((3, -9)\).

This vertex tells us where the parabola changes direction, making it a key element in graphing the function.

X-intercepts are the points where the graph of a function crosses the x-axis. These points indicate the values of \( x \) for which the function \( f(x) \) becomes zero. To find the x-intercepts of a quadratic function, set \( f(x) = 0 \) and solve for \( x \).

For the function \( f(x) = x^2 - 6x \):

• Set \( x^2 - 6x = 0 \).
• Factor the equation: \( x(x - 6) = 0 \).
• So, \( x = 0 \) or \( x = 6 \).

The x-intercepts are \((0, 0)\) and \((6, 0)\).

These intercepts are vital for drawing the graph accurately as they show where the parabola intersects the x-axis, allowing us to understand its position and orientation.

Y-intercepts occur where the graph crosses the y-axis. This happens when \( x = 0 \). Essentially, you substitute zero into the function to find the y-intercept. For every quadratic equation written in standard form, the y-intercept is simply the constant term, \( c \).

In our example, \( f(x) = x^2 - 6x \), substitute \( x = 0 \):

• \( f(0) = 0^2 - 6 \times 0 = 0 \)

The y-intercept is \((0, 0)\), which also serves as one of the x-intercepts. This point indicates where the parabola will touch or cross the y-axis, providing a reference point for graphing the function.

The standard form of a quadratic equation is crucial for understanding and analyzing the function's properties. It's given by \( f(x) = ax^2 + bx + c \), where \( a \), \( b \), and \( c \) are constants. This format not only helps in identifying the nature of the parabola but also in finding the vertex, x-intercepts, and y-intercepts easily.

For the quadratic function \( f(x) = x^2 - 6x \):

• The equation is already in standard form with \( a = 1 \), \( b = -6 \), and \( c = 0 \).
• This allows us to easily calculate the vertex and intercepts using standard methods.

The sign of \( a \) determines if the parabola opens upwards \((a > 0)\) or downwards \((a < 0)\). In this case, the parabola opens upwards.

Graphing parabolas requires understanding their key features, such as the vertex, intercepts, and symmetry. A parabola opens upwards or downwards depending on the coefficient \( a \) in the quadratic equation \( f(x) = ax^2 + bx + c \).

To graph the function \( f(x) = x^2 - 6x \):

• Identify the vertex \((3, -9)\), which is the lowest point in this upward-opening parabola.
• Mark the x-intercepts \((0, 0)\) and \((6, 0)\) on the graph, indicating where the parabola crosses the x-axis.
• Since the y-intercept is \((0, 0)\), this point will also aid in shaping the parabola graph.
• The axis of symmetry, \( x = 3 \), splits the parabola into two mirrored halves.

The resulting graph will be a smooth curve that hugs the lowest point at the vertex and passes through the intercepts, giving a clear visual of the function's behavior.