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When dealing with quadratic functions, it's important to understand their basic form. A quadratic function is a type of polynomial function that takes the shape of a parabola when graphed. These functions can be written in the general form as \( f(x) = ax^2 + bx + c \). Here, \( a \), \( b \), and \( c \) are constants, and \( x \) represents the variable. Quadratic functions are unique because they create symmetrical curves that open either upwards or downwards. This opening direction is dictated by the coefficient \( a \): if \( a \) is positive, the parabola opens upwards; if negative, it opens downwards. This shape helps in determining specific points of interest on the graph, such as the vertex and the axis of symmetry.

The standard form of a quadratic function is expressed as \( f(x) = ax^2 + bx + c \). This is a crucial format for identifying the coefficients which play a key role in solving quadratic equations and finding the vertex of a parabola. The standard form is organized to highlight the quadratic coefficient \( a \), the linear coefficient \( b \), and the constant term \( c \). Recognizing these coefficients is essential because they help determine the properties of the parabola, such as the orientation and position on the Cartesian plane.

• Orientation: Based on the sign of \( a \), the parabola's direction (upward or downward) is deduced.
• Vertex Calculation: The coefficients are plugged into formulas to calculate the vertex.

Understanding the standard form also lays the foundation for converting and manipulating quadratic equations into other useful forms.

To find the vertex of a parabola described by a quadratic function in standard form \( ax^2 + bx + c \), we use the vertex formula. The vertex is represented by the coordinates \((h, k)\). The formula for calculating \( h \), the x-coordinate of the vertex, is given by:\[ h = -\frac{b}{2a} \]This formula results from identifying the axis of symmetry of the parabola. Once \( h \) is found, we substitute it back into the original quadratic function to find \( k \), the y-coordinate:\[ k = f(h) \]Understanding this formula is crucial because it allows us to directly calculate the vertex, a significant point that acts as the peak or trough of the parabola depending on its orientation.

• This formula simplifies finding the vertex without needing to complete the square, a process that can sometimes be cumbersome.
• It's a quick computational solution for determining the most important feature of a parabola.

The process of coordinate calculation involves determining the x and y values for the vertex of a parabola. Let's discuss how these are calculated step-by-step using the given quadratic function. First, to find the x-coordinate \( h \), apply the vertex formula:\[ h = -\frac{b}{2a} \]In our example, substituting the constants \( a = 0.05 \) and \( b = 2.5 \) provides:\[ h = -\frac{2.5}{2 \times 0.05} = -25 \]Next, calculate the y-coordinate \( k \) by substituting \( h \) back into the quadratic function \( f(x) = 0.05x^2 + 2.5x - 1.5 \):\[ k = 0.05(-25)^2 + 2.5(-25) - 1.5 \]Proceed with the calculations:

• \((-25)^2 = 625\)
• \(0.05 \times 625 = 31.25\)
• \(2.5 \times (-25) = -62.5\)

Finally, sum these numbers to get \( k = 31.25 - 62.5 - 1.5 = -32.75 \).Thus, the vertex coordinates are \((-25, -32.75)\), characterizing the turning point of the quadratic function.