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A quadratic function is a type of polynomial function, specifically one that can be written in the standard form: \( f(x) = ax^2 + bx + c \). Here, \(a\), \(b\), and \(c\) are constants, with \(a eq 0\). The graph of a quadratic function is a curve called a parabola. Parabolas have some interesting properties, such as their reflective symmetry and the fact that they open upwards if \(a > 0\) or downwards if \(a < 0\). This orientation depends on the coefficient \(a\).To better understand quadratic functions, let's look at their properties:

• Vertex: The vertex form of a quadratic function is useful to quickly identify the vertex. The vertex is the highest or lowest point on the graph, depending on whether the parabola opens downwards or upwards.
• Axis of symmetry: This is a vertical line that passes through the vertex and divides the parabola into mirror images of each other.
• Roots or x-intercepts: These are the points where the graph intersects the x-axis. They are found by solving \(f(x) = 0\).

A simple example is the function \( f(x) = -2x^2 + 8x - 10 \), which opens downwards since \( a = -2 \) is negative.

The x-intercepts of a function are the points where the graph crosses the x-axis. For a quadratic function in the form \( f(x) = ax^2 + bx + c \), finding the x-intercepts involves setting \( f(x) = 0 \) and solving for \( x \).The solution to this equation can be obtained using the quadratic formula: \[ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \] where \(a\), \(b\), and \(c\) are coefficients from the quadratic equation.

• **Real roots:** If the discriminant \(b^2 - 4ac\) is positive, the function will have two distinct real x-intercepts.
• **One real root:** If the discriminant is zero, there will be exactly one real x-intercept (the graph touches the x-axis).
• **No real roots:** If the discriminant is negative, there are no real x-intercepts, which means the parabola doesn't cross the x-axis.

In our example, the discriminant was found to be negative (\( 64 - 80 = -16 \)), indicating no real x-intercepts for the quadratic \( f(x) = -2x^2 + 8x - 10 \). This signifies that the parabola does not intersect the x-axis at any real point.

The y-intercept of a graph is the point where the graph intersects the y-axis. For any function, including quadratics, the y-intercept can be found by evaluating the function at \( x = 0 \).To find the y-intercept of a quadratic function \( f(x) = ax^2 + bx + c \), simply calculate \( f(0) \):- Substitute \(x = 0\) into the quadratic equation: \( f(0) = a(0)^2 + b(0) + c = c \). Therefore, the y-intercept is at \((0, c)\).In the example equation \( f(x) = -2x^2 + 8x - 10 \), the y-intercept is easily obtained:

• The coefficient \(c\) is \(-10\), which means the graph intersects the y-axis at \((0, -10)\).

Thus, the y-intercept is a straightforward way to get insight into where a quadratic graph crosses the y-coordinate. It's also a crucial piece of information for sketching and understanding the general position of the parabola on the coordinate plane. Understanding y-intercepts allows you to start graphing a quadratic function and see where it relates to the origin.