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The vertex formula is a crucial tool in determining the vertex of a quadratic function. A quadratic function is typically expressed in the form:

• \( f(x) = ax^2 + bx + c \)

The vertex of the parabola represented by a quadratic function gives us the highest or lowest point on its graph. To find the x-coordinate of this vertex, we use the formula:

• \( x = -\frac{b}{2a} \)

This formula involves the coefficients \( b \) and \( a \) from the equation. In our example, we have a quadratic function \( f(x) = x^2 - 40x + 500 \). Here, the coefficients are:

• \( a = 1 \)
• \( b = -40 \)

Substituting these values into the vertex formula provides us the x-coordinate of the vertex:

• \( x = -\frac{-40}{2 \times 1} = 20 \)

After finding the x-coordinate, plug it back into the original function to find the y-coordinate. For \( x = 20 \), \( f(20) = 100 \). Thus, the vertex is \( (20, 100) \). This point is pivotal as it helps in sketching out the parabola accurately.

Graphing a parabola involves utilizing the vertex and several other points to properly sketch its shape on a coordinate plane. When graphing the quadratic function \( f(x) = x^2 - 40x + 500 \), the first step is to identify the vertex, which we've calculated as \( (20, 100) \). Since the coefficient \( a \) is positive, the parabola opens upwards. An upwards opening parabola implies that as \( x \) moves away from the vertex, the \( y \)-values increase. To sketch the graph:

• Start by marking the vertex \( (20, 100) \) on the graph.
• Select additional x-values around 20, like 18, 19, 21, and 22.
• Calculate their corresponding \( y \)-values using the quadratic function to obtain other points on the graph.
• Plot these points on your graph paper to ensure accuracy.
• Draw a smooth curve through all the points, emphasizing the parabola's symmetry around the vertex axis.

By following this step-by-step process, you will have an accurate representation of the quadratic function as a parabola on the graph.

Quadratic coefficients play a fundamental role in determining the characteristics of a quadratic function. In the expression \( f(x) = ax^2 + bx + c \), the values of \( a \), \( b \), and \( c \) determine:

• The direction in which the parabola opens.
• The steepness or narrowness of the parabola.
• The position of the parabola on the coordinate plane.

The coefficient \( a \) impacts how the graph stretches or compresses. A larger absolute value of \( a \) results in a steeper curve. Conversely, when \( a \) is smaller but still positive, the parabola appears wider. The sign of \( a \) is also crucial:

• A positive \( a \) means the parabola opens upwards.
• A negative \( a \) indicates it opens downwards.

In our example, \( a = 1 \), so the parabola opens upwards and is relatively standard in width.The coefficient \( b \) influences the vertex’s horizontal position, while \( c \) shifts the function vertically on the graph. For \( f(x) = x^2 - 40x + 500 \):

• \( b = -40 \) affects where the vertex lies along the x-axis.
• \( c = 500 \) sets the starting height of the parabola.

Coherently analyzing these coefficients allows for a clearer understanding and precise graphing of any quadratic function.