(finding the equation of a parabola)

In this section we learn how to find the equation of a parabola, using root factoring.

Given the graph a parabola such that we know the value of:

• its two $$x$$-intercepts (the two points at which the parabola cuts the $$x$$-axis), or
• its single $$x$$-intercept, if the parabola only cuts the $$x$$-axis once
and we know the coordinates of one other point through which the parabola passes then we can find the parabola's equation using root factoring.

By the end of this section we'll know how to find the equation of any of the parabola shown here:

We start by learning how to write a parabola's equation in root factored form when the parabola has two $$x$$-intercepts as well as watch a couple of detailed tutorials showing us how this can be used to find a parabola's equation.

We'll then learn about the scenario in which the parabola has one $$x$$-intercept, which we'll also illustrate with a detailed tutorial.

Root Factored Form: two $$x$$-intercepts

If a parabola cuts the $$x$$-axis in two points: $\begin{pmatrix}p,0\end{pmatrix} \quad \text{and} \quad \begin{pmatrix}q,0\end{pmatrix}$ then the parabola's equation, $$y=ax^2+bx+c$$, can be written: $y = a\begin{pmatrix}x - p \end{pmatrix}\begin{pmatrix}x - q \end{pmatrix}$ where $$p$$ and $$q$$ are the $$x$$-coordinates of the points at which the parabola cuts the $$x$$-axis (the $$x$$-intercepts).

Finding a Parabola's Equation

Given the graph of a parabola, like either of the two shown here, we can use root factoring to find their parabola's equations.

Note: notice that in each these graphs we can see both:

• the $$x$$-intercepts, and
• the coordinates of one other point the curve passes through.

The method for finding a parabola's equation, when given a graph like those we have here, is clearly explained in the following tutorials, make sure to watch them now.

Tutorial 1

In this first tutorial we learn how to find the equation of a parabola given:

• it cuts the $$x$$-axis in two points, that we are given or can read from the graph
• we know the value of the parabola's $$y$$-intercept.

Tutorial 2

In this second tutorial we learn how to find the equation of a parabola given:

• it cuts the $$x$$-axis in two points, that we are given or can read from the graph
• we know the coordinates of one other point through which the parabola passes (other than the $$y$$-intercept).

Exercise 1

Use root factoring to find the equation of each of the parabola shown below.

In each case, write the parabola's equation in root factored form and in the general $$y=ax^2+bx+c$$ form.

1. We find: $y=2(x-1)(x-4)$
and so:
$y = 2x^2-10x+8$
2. We find: $y=-(x+2)(x-4)$
and so:
$y = -x^2+2x+8$
3. We find: $y=3(x+3)(x-1)$
and so:
$y = 3x^2+6x-9$
4. We find: $y=2(x-1)(x-4)$
and so:
$y = 2x^2-10x+8$
5. We find: $y=(x+2)(x-6)$
and so:
$y = x^2-4x-12$
6. We find: $y=-2(x+1)(x-3)$
and so:
$y = -2x^2+4x+6$

Root Factored Form: one $$x$$-intercept

If a parabola cuts the $$x$$-axis in one point: $\begin{pmatrix}p,0\end{pmatrix}$ then the parabola's equation, $$y=ax^2+bx+c$$, can be written: $y = a\begin{pmatrix}x - p \end{pmatrix}^2$ where $$p$$ is the $$x$$-coordinate of the point at which the parabola cuts the $$x$$-axis (the $$x$$-intercept).

Tutorial 3

In this third tutorial we learn how to find the equation of a parabola given:

• it cuts/tcouhes the $$x$$-axis in only one point
• we know the coordinates of one other point through which the parabola passes.

Exercise 2

Use root factoring to find the equation of each of the parabola shown below.

In each case, write the parabola's equation in root factored form and in the general $$y=ax^2+bx+c$$ form.

1. We find: $y=5(x-1)^2$
and so:
$y = 5x^2-10x+5$
2. We find: $y=-2(x-2)^2$
and so:
$y = -2x^2+8x-8$
3. We find: $y=(x+3)^2$
and so:
$y = x^2+6x+9$
4. We find: $y=4(x+1)^2$
and so:
$y = 4x^2+8x+4$
5. We find: $y=3(x-2)^2$
and so:
$y = 3x^2-12x+12$
6. We find: $y=-2(x-1)^2$
and so:
$y = -2x^2+4x-2$