Parabola

Refer to math is fun: Parabola : The graph for quadratic

Parabola is a quadratic's graph.

A parabola always has a vertex, either its top point or bottom point.
If you draw a vertical line goes through the vertex, it can divided the graph into two mirrored part.

The Concavity of parabola

• When it opens up, we also call it Concave up,
• When it opens down, we also call it Concave down.

Equation forms of a Parabola

Refer to math warehouse: Equation forms of a Parabola

There’re two common forms of equation to express the parabola:
Standard form and Vertex form.

And there’s another form for the parabola:
the Factor form.

How to get the Axis of symmetry

Refer to math warehouse: How to get the Axis of symmetry

For a parabola, the Axis of symmetry goes through the vertex, which represent the x position of the vertex.

The formula equation of Axis of symmetry:

• In the Vertex form of quadratic y = (x-h)² + k, the symmetry line is: x = h.

Mind that, to understand this form, refers to the Absolute value graph notion.

• In the Standard form of quadratic y=ax²+bx+c, the symmetry line is: x=−b/2a

• In the Factor form of quadratic y=c(x-a)(x-b), the symmetry line is: x=(a+b)/2

How to get the Vertex

Getting the Axis of symmetry is half way to get the vertex, since it can only represent the x position of vertex.
For getting the y position, just input the x value into the equation, and get the y value.
Solved.

It’s easier to get the vertex in the factor form of quadratic:

Then the vertex is (h, k).

How to graph the quadratic

To graph a quadratic, we have a few different ways, and each needs the information of a few points:

• A vertex point, two root points.
• A vertex point, a y-intersect point.
• A vertex point, any two points on the graph.

Refer this page to review how to get a parabola’s intersects.

• To get two roots:

Just solve the equation and get two solutons (if there’re two solutions).

• To get the y-intersect point:

Simply let the x=0,
and solve the quadratic to get y position of the vertex.
And input the y value to the equation, to solve x position. Then we get the vertex:
(x, y)

• To get any two points of the graph:

Just ASSUME two x positions,
then input to the quadratic, to get y positions. Then we get:
(x1, y1) and (x2, y2)

Parabola from geometric perspective

Refer to khan academy: Parabola from geometric perspective

Definition: A parabola is the set of ALL POINTS whose distance from a certain POINT (the focus) is equal to their distance from a certain LINE (the directrix).

It means,

To draw a parabola, you only need a point and a line.

Khan lecture.
Maths if fun.

Equation for the parabola

In this graph above, we will get an equation by Pythagorean Theorem

And expand that equation, we got a parabola equation in Vertex quadratic form:

y = c (x - i)² + j

So we could spot out the focus and directrix information from a vertex quadratic equation.
In which, the (i, j) represent the vertex, (h, k) represents the focus, and the y=k represents the directrix.

Equation of a parabola from focus & directrix

Khan practice.

Example

Solve:
According to the focus and directrix information, we know we can consist an equation like:

(y-k)² = (x-a)² + (y-b)²

But that’s for a vertical parabola. When the directrix is obviously making it to a horizontal parabola, we have to change the equation to:

(x-k)² = (x-a)² + (y-b)²

So it becomes:

(x+7)² = (x+3)² + (y+5)²

Expand the equation to get: