91Ó°ÊÓ

The quadratic formula is a powerful tool for finding the solutions, or "zeros," of quadratic equations. A quadratic equation has the standard form \( ax^2 + bx + c = 0 \), where \( a, b, \) and \( c \) are constants. The formula is given by:\[ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \]This formula is derived from the process of completing the square and provides both roots of the equation. When solving a quadratic equation, you plug in the values of \( a, b, \) and \( c \) into the formula to find the solutions for \( x \).

• The expression \( b^2 - 4ac \) is called the "discriminant" and determines the nature of the roots.
• If the discriminant is positive, there are two distinct real roots.
• If it is zero, there is exactly one real root (a repeated root).
• If negative, the roots are complex numbers.

In our given problem, we have \( a = -2 \), \( b = 20 \), and \( c = -43 \). Plugging these into the quadratic formula gives us the zeros of the function.

The zeros of a quadratic function are the values of \( x \) that make the function equal to zero, i.e., \( f(x) = 0 \). For a quadratic function represented as \( f(x) = ax^2 + bx + c \), the zeros are where the parabola crosses or touches the x-axis. In the problem we have, after substituting the coefficient values into the quadratic formula, we find the zeros to be:\[ x = 5 + \frac{\sqrt{14}}{2} \quad \text{and} \quad x = 5 - \frac{\sqrt{14}}{2} \]These zeros are crucial in graphing the parabola, as they provide the x-intercepts.

• Zeros can also be called "roots" or "solutions" of the equation.
• They are the points where the graph will touch the x-axis.
• Knowing these helps us understand the shape and position of the graph.

The vertex of a parabola is either its highest or lowest point, depending on whether the parabola opens upwards or downwards. For a quadratic function \( f(x) = ax^2 + bx + c \), we can find the x-value of the vertex using the formula:\[ x = -\frac{b}{2a} \]For our function, substituting \( a = -2 \) and \( b = 20 \) gives:\[ x = -\frac{20}{2(-2)} = 5 \]Replace \( x = 5 \) back into the function to find the y-coordinate of the vertex, which gives us \( f(5) = 7 \). Therefore, the vertex is at \((5, 7)\).

• If \( a > 0 \), the parabola opens upwards, and the vertex is the minimum point.
• If \( a < 0 \), the parabola opens downwards, and the vertex is the maximum point.
• In our case, the parabola opens downwards, and the vertex is a maximum point since \( a = -2 \).

Graphing a quadratic function involves plotting its key features: the vertex, the axis of symmetry, and the zeros or x-intercepts, if they exist. For a function \( f(x) = ax^2 + bx + c \), these points help visualize the shape and direction of the parabola.

• The vertex represents the maximum or minimum point and provides a point of symmetry for the graph.
• The axis of symmetry can be found at \( x = -\frac{b}{2a} \) and crosses through the vertex.
• The zeros \( x = \) where \( f(x) = 0 \) are points where the parabola will intersect the x-axis.

For our quadratic function, the vertex is \((5, 7)\), and it opens downwards due to the negative coefficient \( a = -2 \). The parabola will intersect the x-axis at the calculated zeros. Plotting additional points by substituting x-values near the vertex can also assist in accurately sketching the parabola. Consider symmetry, and remember the parabolic curve balances equally over the axis of symmetry, creating its distinct U-shape or inverted U-shape.