The vertex of a parabola or quadratic equation is the curve’s highest or the lowest point and is also called the maximum or the minimum point of a curve. The vertex can be found on the line of symmetry of a parabola. The line of symmetry divides the figure into two halves and both are mirror images.
To find the vertex:
Using the formula method: The formula to find the vertex is:
f (x) = ax2 + bx + c
If you have a quadratic equation y=x2+ 9x + 18
When you replace the quadratic equation with the values of the vertex formula, then x2= a, x= b and the constant = c. Using the above equation y=x2+ 9x + 18 and replacing the values a=1, b= 9 and c= 18.
To find the value of x: The axis of symmetry is the vertex and using the formula to find the value of x, x = -b/2a. Replace the values of x and also the values of a and b to get
Find the value of y: Now that you have derived the value of x, it is now easy to find the value of y by replacing the x value into the formula.
y = x2 + 9x + 18
y = (-9/2)2 + 9(-9/2) +18
y = 81/4 -81/2 + 18
y = 81/4 -162/4 + 72/4
y = (81 – 162 + 72)/4
y = -9/4
Now that you have both the values of x and y, it can now be written in an ordered pair (x,y) or (-9/2, -9/4) and this becomes the vertex of the quadratic equation. If the same is plotted on a graph, the point (-9/2, -9/4) would be a minimum as x2 is positive.
Completing the square method: This is another method to find the vertex of a quadratic equation. The equation for this form will be y = ax2 + bx + c.
To find the vertex of y = 2×2 – 4x + 5
Since ‘completing the square’ method is being used the x2 and x terms will be moved to one side, and +5 will be moved to the left of the = sign. So the equation now becomes
y – 5 = 2×2 – 4x
A coefficient of 1 is needed to complete the square, and so 2 has to be factored out
y – 5 = 2(x2 – 2x)
Next step is to find the perfect square.To do that half the coefficient of the x inside the bracket, square it and place it in the equation which becomes y -5+2 = 2(x2 -2x +1)
Converting the right part of the equation to a square becomes 2(x-1)2
The equation now becomes y-3 = 2(x-1)2
To find the value of y: y = 2(x-1)2+3
The vertex (x,y) is (1,3) which can now be plotted in the graph.
Example of a real-life application of a vertex: Throwing a ball across the room you will see that there is an arc shape to it as the ball goes high and then falls into the hands of the catcher. The ball here has travelled both horizontally as well as vertically. The path the ball traverses is a parabola, and when you throw it and the ball reaches its high it is the parabola’s maximum and when it lands it is it’s minimum.