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Quadratic functions expressed in the standard form are a key building block in understanding their behavior and characteristics. The standard form of a quadratic function is given by \( f(x) = ax^2 + bx + c \), where:

• \( a \) determines the opening direction and the width of the parabola.
• \( b \) affects the position of the vertex.
• \( c \) represents the \( y \)-intercept of the parabola.

For example, the function \( f(x) = x^2 - 6x \) is already in standard form. Here, the coefficients are \( a = 1 \), \( b = -6 \), and \( c = 0 \). This makes it simple to apply other algebraic methods to find more features of the graph. The coefficient \( a = 1 \) indicates that the parabola opens upwards and is relatively narrow.

The vertex of a quadratic function provides us with the maximum or minimum point of the graph, depending on whether the parabola opens upwards or downwards. The vertex can be found using the formula:\[x = -\frac{b}{2a}\]After finding \( x \), substitute it back into the function to find the \( y \)-value of the vertex.
For example, in the function \( f(x) = x^2 - 6x \), the value for \( x \) is calculated as:\[x = -\frac{-6}{2 \times 1} = 3\]Substitute \( x = 3 \) into the original function to find the \( y \)-value:\[f(3) = 3^2 - 6 \times 3 = 9 - 18 = -9\]Thus, the vertex of this function is at the point \((3, -9)\). This tells us the lowest point (minimum) on the graph because the parabola opens upwards.

Intercepts are crucial points where the graph intersects the axes and they provide insightful information about the quadratic function. There are two types of intercepts:

• X-intercepts: Points where the graph crosses the \(x\)-axis. Found by solving the equation \( f(x) = 0 \).
• Y-intercept: The point where the graph crosses the \(y\)-axis. Found by evaluating the function at \( x = 0 \).

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

• The \( x \)-intercepts are found by solving the equation \( x^2 - 6x = 0 \), which factors to \( x(x - 6) = 0 \). Thus, the x-intercepts are \( x = 0 \) and \( x = 6 \).
• The \( y \)-intercept is simply \( c = 0 \), so the \( y \)-intercept is at \( (0, 0) \).

Sketching a parabola involves plotting significant points such as the vertex and intercepts and understanding the parabola's symmetry and direction.Here's a simple way to sketch the parabola:

• Plot the vertex, which in our example is \((3, -9)\).
• Mark the \( x \)-intercepts at \((0, 0)\) and \((6, 0)\).
• Indicate the \( y \)-intercept located at \((0, 0)\).
• Draw the parabola opening upwards through these points, creating the U-shape characteristic of quadratic functions.
• The axis of symmetry passes through the vertex, providing a mirror line, which is \( x = 3 \) in this example.

These steps form the basis of graphing any quadratic function, ensuring that the major points and the symmetry of the parabola are accurately represented.