Parabola Equation

**nath_quam** · June 9th 2006, 07:56 PM

Is there an equation for a parabola which has a vertex at (0,1) and goes through the point (1,3)

**earboth** · June 9th 2006, 08:20 PM

Originally Posted by nath_quam

Is there an equation for a parabola which has a vertex at (0,1) and goes through the point (1,3)

Hello,

if the vertex is $V(x_v \vert y_v)$ then you can use the vertex-form of the parabola:
$p(x)=a\cdot (x-x_v)^2+y_v$

Plug in the coordinates of the vertex:

$p(x)=a\cdot (x-0)^2+1$ . Now is left only the coefficient a. Plug in the coordinates of the point (I've named it A):

$3=a\cdot (1-0)^2+1$ and you'll get a = 2.

So the parabola is: $p(x)=2\cdot (x)^2+1$

I've attached a diagram.

Greetings

EB

**Rich B.** · June 9th 2006, 08:31 PM

nath_quamIs there an equation for a parabola which has a vertex at (0,1) and goes through the point (1,3)

Hi Nath_Quam: In short, yes, there are infinitely many such parabolas that meet the conditions given in your narrative. To begin, we write your parabola in functional notation, i.e., f(x) = ax^2 + bx + c. [Note: There is nothing compelling us to use f(x). We could just as legitimately have written g(x)=..., h(x)=..., or, simply y= ax^2 + bx + c. All are quadratic equations and, hence, have parabolic graphical representations].

Now, recall that any parabola has x-value: -b/2a. According to your given data, the vertex is at (1,3). Hence 1 =-b/2a, and we immediately observe b=0 in function, f(x) = ax^2 + bx + c. So, f(x) = ax^2+c.

**earboth** · June 9th 2006, 08:37 PM

Originally Posted by Rich B.

nath_quamIs there an equation for a parabola which has a vertex at (0,1) and goes through the point (1,3)
...
Now, recall that any parabola has x-value: -b/2a. According to your given data, the vertex is at (1,3). Hence 1 =-b/2a, and we immediately observe b=0 in function, f(x) = ax^2 + bx + c. So, f(x) = ax^2+c.

Hello,

I understood that the vertex is V(0,1). Did I miss something?

Greetings

EB

**Soroban** · June 10th 2006, 03:43 AM

Hello, nath_quam!

Is there an equation for a parabola which has a vertex at (0,1) and goes through the point (1,3)

Assuming it is a "vertical" parabola, it has the form: . $y\:=\:ax^2 + bx + c$

The vertex of a parabola is at: $x = \frac{-b}{2a}$

Since this parabola has its vertex at $x = 0$ , we see that $b = 0.$

. . The equation (so far) is: . $y\:=\:ax^2 + c$

Since $(0,1)$ is on the parabola: . $1\:=\:a\cdot0^2 + c\quad\Rightarrow\quad c = 1$

. . The equation is: . $y\:=\:ax^2 + 1$

Since $(1,3)$ is on the parabola: . $3\:=\:a\cdot1^2 + 1\quad\Rightarrow\quad a = 2$

. . Therefore, the equation is: . $\boxed{y \:=\:2x^2 + 1}$

**Rich B.** · June 10th 2006, 05:02 AM

Originally Posted by earboth

Hello,

I understood that the vertex is V(0,1). Did I miss something?

Greetings

EB

Greetings Earboth:

Late last night, I began the process of addressing the given inquiry. After completing a few lines, however, circumstances arose such that I was unable to finish that which I had begun. I thought I had clicked the "cancel" icon but...

This morning, I awoke to read two well articulated, mathematically complete responses to the originator's query. Any contributory words I may have to offer now, would be an inefficient streatch of my stiff, early morning, pre-caffine enhanced cerebrum.

And that's that. Some few hours from now, I shall delete the initial post, as well as this clarification of the same.

Enjoy your day.

Regards,

Rich B.

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