How do we Understand the Vertex Formula?
Assume that you throw a ball from the ground at some height. The ball makes some curve and then falls on the ground. Now, here, we say that the path a ball took is Parabola. Please note that here we talked about Parabola, which is symmetrical.
Now, if we plot a graph having x and y-axis, we see a point where this parabola (or the path that a ball took) crosses the axis of symmetry is actually the vertex of the parabola. Let’s learn this using the vertex formula.
Assume that a point (m, n) cuts the x-axis at some coordinates, we call them vertices. So, here we have the following Parabola standard equation:
y = ap2 + bp + c
So, we notice that the path of a ball cuts its symmetric axis at the x-axis, so we get our equation as;
y = a (p – m)2 + n….(2)
Here, eq (2) is the required vertex form of the standard equation y = ap2 + bp + c.
Story of a Hodophile Niya | Parabola – Vertex Formula
Niya loves to take a U-turn of every path she takes, for it is riding a car at one end of the road, then taking a U-turn and coming back to the same path. It also happens that she goes on mount-trekking to Kanchenjunga peak, like once she has reached the peak, and then goes back to the ground and makes a U-turn to reach back to the peak. Wow! Such an energetic girl, Niya is!
But let’s wait and see what we are discussing and what is the use of talking about it? Well, in mathematics, this U-turn is actually Parabola. So, what parabola is?
Well, Parabola is a plane curve that is U-shaped just like Niya’s traveling style and this curve is mirror-symmetrical as well. Here, we noticed another thing that when Niya started her journey from point A, reached some distance ‘x km’ and retraced her path, she came back to A’. We see that if we draw a mirror in between, we notice that A and A’ are mirror images lying at the same distance, this is what happens in Parabola.
Now, the very point, Niya took U-turn is actually the mirror kept between A and A’ points, and this is the point, where Parabola cuts its symmetrical axis, which we call the vertex. We do have the vertex formula for the same. Here, we will understand this formula in simple words.
Another Vertex Form for Parabola
So, friends, we can also write the vertex form of Parabola in another way, which is:
Now, let us understand the ways to determine coordinates (m, n) of the vertex:
- (m, n) = (- b/2a, – D/4a), when D = b2 – 4ac
- When m = – b / 2a, we need to evaluate y at m to find n.
Simplest Example on Standard Form to Vertex Form
Now, let us take an equation: y = a2 + 6a – 5 and make it a vertex form:
Add and subtract 32 in the above equation:
a2 + 6a – 5 + 32 – 32
a2 + 6a + 32 – 5 – 32
(a + 3)2 – 14
So, the roots of this equation are (- 3, 14)
Now, let us use vertex formula:
m = – b/2a = – 6/2.1 = – 3, and
n = we will calculate this part in the following two parts:
First: b2 – 4ac = (62 – 4. 1. (- 5)) = 56
Second: 1/4a = 1/4. 1 = 14
This verifies that coordinates of the vertex are (- 3, 14).
Another Solved Example on Vertex Formula
Suppose we have an equation: y = 3x2 – 5x + 2. So, find the x-vertex and y-vertex for this parabola.
Here, a = 3, b = – 5, and c = 2
x-vertex = – b/2a = – (-5)/2. 3 = ⅚
y-vertex = (b2 – 4ac)/4a, we will calculate this part in the following two parts:
First: b2 – 4ac = (- 5)2 – 4. 3. 2 = 1
Second: 1/4a = 1/4. 2 = ⅛
So, the coordinates of vertex or at the point the parabola cuts its symmetric axis (m, n) are (⅚, ⅛). This is how we calculate the vertex coordinate of the Parabola.
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