Introduction

A Quadratic function, or parabola, is any function of the form ${\displaystyle f(x)=ax^{2}+bx+c}$. Quadratic functions have either a unique maximum or a unique minimum. The point where the maximum or minimum is reached is called the vertex.

Definition and properties

Even if we know the form of a quadratic function it is not very useful because we do not have any useful information immediately available to us. This first formula gives us an equivalent function that provides more immediate information.

If ${\displaystyle f(x)=ax^{2}+bx+c}$, then we can set ${\displaystyle h=-{\frac {b}{2a}}{\text{ and }}k={\frac {4ac-b^{2}}{4a}}}$ then we have that ${\displaystyle f(x)=a(x-h)^{2}+k}$ Now it is more obvious that f(x) is the parabola ${\displaystyle y=ax^{2}}$ shifted horizontally by h units and vertically by k units. The parabola opens up(down) if a is positive(negative). The maximum or minimum occurs at (h, k), also known as the vertex, and the axis of symmetry is the vertical line x = h.

To summarize: Given a quadratic equation ${\displaystyle f(x)=ax^{2}+bx+c}$, the vertex is the point ${\displaystyle \left(-{\frac {b}{2a}},f\left(-{\frac {b}{2a}}\right)\right)}$ and the axis of symmetry is the vertical line ${\displaystyle x=-{\frac {b}{2a}}}$

Discriminant and x-intercepts of a Quadratic Function

Given a quadratic equation ${\displaystyle ax^{2}+bx+c}$ the discriminant is ${\displaystyle b^{2}-4ac}$. You may recognize this from the quadratic formula. We will use the relation between the discriminant and quadratic formula to discuss the x-intercepts of the quadratic function

1) If the discriminant is positive the quadratic function has two distinct x-intercepts

2) If the discriminant is zero the quadratic function has one x-intercept at the vertex

3) If the discriminant is negative the quadratic function has no x-intercepts.

Determining the Quadratic Function from its Vertex and One Other Point

Since we know a quadratic function can be written in the form ${\displaystyle f(x)=a(x-h)^{2}+k}$, we can determine the quadratic equation going through two points, if one of them is the vertex. This achievable since (h, k) is the vertex of the quadratic function.

Example: Find the quadratic function with vertex (1, 3) going through (3, 0).

Since the vertex is (1, 3) we know ${\displaystyle f(x)=a(x-1)^{2}+3}$. Now we just need to solve for a. To do this we plug in the other point. So ${\displaystyle 0=a(3-1)^{2}+3}$ Solving this equation we find that ${\displaystyle a=-{\frac {3}{4}}}$