Lesson : : Quadratic Funcitons

A quadratic function is a type of polynomial function that specifically follows a second-degree polynomial equation.

The general form of a quadratic function is \( f(x) = ax^2 + bx + c \).

1. The parabola opens upwards.

2. The vertex (the lowest point of the curve if it opens upward) is the minimum point on the graph

3. It intersects the y-axis at the point where, called the y-intercept.

4. It has no x-intercepts if the vertex is above the x-axis (if the discriminant b2−4ac is negative), or it has two x-intercepts (real roots) if the discriminant is positive.

1. The parabola opens downwards.

2. The vertex (the highest point of the curve if it opens downward) is the maximum point on the        graph.

3. It intersects the y-axis at the point where $x=0$, called the y-intercept.

4. It behaves similarly to the case where $a>0$ but is inverted

A quadratic function in factor form is a way to represent a quadratic equation by factoring it into its constituent linear factors. In this form, the quadratic equation is expressed as the product of two linear factors.

The general form of a quadratic function in factors form is \( f(x) = a(x – r)(x – s) \)

Where:

• • • f(x) represents the function value at x.
    • x is the independent variable.
    • a is the leading coefficient that scales the entire function.
    • (r, s) are the roots or zeros of the quadratic equation, which are the x-values where the function equals zero. 

A quadratic function in vertex form is a way to express a quadratic equation that highlights the vertex of the parabolic graph.

The vertex form of a quadratic function is given by \( f(x) = a(x-h)^2+k\)

Where:

• • • f(x) represents the function value at x.
    • x is the independent variable.
    • a is the coefficient that determines the direction (opening) and scale of the parabola. If “a” is positive, the parabola opens upward; if “a” is negative, the parabola opens downward.
    • (h, k) represents the coordinates of the vertex of the parabola. The value “h” determines the horizontal shift of the parabola, and “k” determines the vertical shift.