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A parabola is a U-shaped curve that represents the graph of a quadratic function. Quadratic functions have the form \(f(x) = ax^2 + bx + c\), where \(a\), \(b\), and \(c\) are constants.

• The direction the parabola opens (upward or downward) is determined by the sign of the coefficient \(a\).
• If \(a\) is positive, the parabola opens upwards, forming a 'U' shape.
• If \(a\) is negative, the parabola opens downwards, forming an inverted 'U'.

For example, in the function \(f(x) = x^2 - 4x\), the coefficient of \(x^2\) is 1, which is positive. Therefore, this parabola opens upwards. Understanding how the coefficient of \(x^2\) changes the graph helps in anticipating the general shape of the graph, which is crucial when sketching quadratics.

The quadratic formula is an essential tool for solving quadratic equations. A quadratic equation is typically written as \(ax^2 + bx + c = 0\). The quadratic formula is given by:\[x = \frac{{-b \pm \sqrt{{b^2 - 4ac}}}}{2a}\]To use this formula, it's important to identify the coefficients \(a\), \(b\), and \(c\) from the quadratic equation.
In our example \(f(x) = x^2 - 4x\) rewritten as \(x^2 - 4x + 0 = 0\), we found \(a = 1\), \(b = -4\), and \(c = 0\). These coefficients are substituted into the quadratic formula to find the values of \(x\) that make the function zero.
The formula involves:

• Calculating the discriminant \(b^2 - 4ac\), which determines the nature of the roots (real or complex).
• Using the '+' and '-' symbols in the formula to find two possible solutions for \(x\).

It serves as a universal method for solving any quadratic equation, making it a powerful and versatile mathematical tool.

The zeros of a function are the points where the graph crosses the x-axis. At these points, the value of the function is zero. Finding the zeros of a quadratic function is crucial as they show where the parabola intersects the x-axis.
To find the zeros, use the quadratic formula to solve \(ax^2 + bx + c = 0\). In our example, after substituting in the values, we simplified:\[x = \frac{{4 \pm 4}}{2}\]This resulted in two zeros: \(x = 4\) and \(x = 0\).

• The zeros can be considered as solutions of the quadratic equation.
• They are also called roots or x-intercepts.

Knowing the zeros helps when sketching the graph, as these are key points where the graph will touch or cross the x-axis.

The vertex of a parabola represents the highest or lowest point on the graph. It occurs at the axis of symmetry, a vertical line that passes through the vertex. For a parabola given by the equation \(y = ax^2 + bx + c\), the x-coordinate of the vertex can be found using:\[x = -\frac{b}{2a}\]
In our quadratic function \(f(x) = x^2 - 4x\), we calculated:\[x = -\frac{-4}{2 \cdot 1} = 2\]By substituting \(x = 2\) back into the function, we found the y-coordinate to be \(-4\), making the vertex \((2, -4)\).
The vertex provides important information about the parabola:

• If the parabola opens upwards, the vertex represents the minimum value of the function.
• If the parabola opens downwards, the vertex represents the maximum value.
• The vertex is a critical point for analyzing the function's properties, such as its range and domain.

Thus, the vertex is a pivotal element in sketching and understanding the graphical representation of quadratic functions.