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The quadratic formula is a powerful tool for solving quadratic equations, which are equations of the form \( f(x) = ax^2 + bx + c \). If you want to find the zeros of a quadratic function, meaning the points where its graph crosses the x-axis, the quadratic formula can help. It's expressed as:\[ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\]This formula helps you determine the values of \( x \) when \( f(x) = 0 \). Here's how you can apply it:

• Identify the coefficients \( a \), \( b \), \( c \) from your quadratic function. For example, in \( f(x) = -4x^2 + 4x - 1 \), we have \( a = -4 \), \( b = 4 \), and \( c = -1 \).
• Plug these values into the formula to find the roots, \( x = \frac{-4 \pm \sqrt{16 - 16}}{-8} \), which simplifies to \( x = \frac{1}{2} \). In this case, the discriminant \( \sqrt{b^2 - 4ac} \) is zero, indicating one repeated root.

Using the quadratic formula not only helps find the zeros, but it also gives insight into the nature of the function's roots by examining the discriminant \( (b^2 - 4ac) \). A positive discriminant indicates two real roots, a zero discriminant indicates one real root, and a negative discriminant indicates complex roots.

Finding the vertex of a parabola is crucial, especially when determining whether the parabola has a maximum or minimum value. The vertex can be understood as the peak (maximum) or the trough (minimum) of the graph. For a parabola given as \( ax^2 + bx + c \), the x-coordinate of the vertex \( (h,k) \) can be found using:\[ h = -\frac{b}{2a}\]After calculating \( h \), substitute it back into the function to find \( k \).

• In \( f(x) = -4x^2 + 4x - 1 \), calculate \( h \): \( h = -\frac{4}{2(-4)} = \frac{1}{2} \).
• Use this value to find \( k \): \( k = f\left(\frac{1}{2}\right) = 0 \), making the vertex \( \left(\frac{1}{2}, 0\right) \).

A negative leading coefficient, as \( a = -4 \), implies the parabola opens downwards, and thus has a maximum at the vertex. Here, the maximum value of the function is 0, occurring at \( x = \frac{1}{2} \). Understanding the vertex is essential in graphing quadratics as it serves as a guide for the direction and position of the parabola.

Graphing quadratic functions involves plotting a parabola on a coordinate plane. These functions can be graphed systematically by understanding their key features: the vertex, axis of symmetry, direction, and intercepts.To start, identify the vertex, which is \( \left(\frac{1}{2}, 0\right) \) for the function \( f(x) = -4x^2 + 4x - 1 \). This tells us the highest point on the graph since the parabola opens downward.Several steps are involved in sketching the graph:

• Vertex: Plot the vertex point on the graph.
• Axis of Symmetry: Draw a vertical line through the vertex. The parabola is symmetric about this line.
• Direction: Since \( a = -4 \) is negative, the parabola opens downward.
• Zero: As solved using the quadratic formula, the parabola touches the x-axis at \( x = \frac{1}{2} \). This intersection is both a vertex and a root.

Understanding these aspects will help you correctly draw the parabola on a graph. The vertex shows where the parabola reaches its peak, the axis of symmetry provides balance, and the zeros indicate where the function crosses the x-axis.