what are the coordinates of the turning of the parabola y=x^2+10x?

Answer 1

y = (x+5)^2 -25 Turning point (-5,-25)

Answer 2

There are two ways to calculate the coordinates of the turning point and I shall put them both here:

Method 1: Differentiation

First, we differentiate the equation y = x^2 + 10x

dy/dx = 2x + 10

A turning point has a gradient of 0, so:

dy/dx = 2x+10 = 0

So, x = -10/2 = -5

This is the x-coordinate of the turning point. To find the y-coordinate, we substitute this value of x into the parabola's equation:

y= (-5)^2 + 10(-5)
y= -25

So, the coordinates of the turning point is (-5, -25)

Method 2: Completing the Square

First, we have to convert the equation y = x^2 +10x into a form a(x-h)^2 + k, like so:

x^2 + 10x
= x^2+ 10x +25 -25
= x^2 + 5x + 5x +25 -25
= x(x+5) +5(x+5) -25
= (x+5)^2 -25

This form is very special, as it can tell you the coordinates of the turning point straightaway. The y-coordinate of the turning point is -25, whereas the x-coordinates is -5.

You can deduce the x-coordinate by putting a negative sign before the value that is inside the brackets together with the x. That's why the x-coordinate is -5, NOT 5. This is a very common mistake done by students.

The y-coordinate is the value outside the brackets. You can read it directly, no need for putting a negative sign before the value.

So, the coordinates of the turning point is (-5, -25)

Answer 3

f (x) = x ² + 10 x
f `(x) = 2 x + 10 = 0 for turning points
x = - 5
Turning point is ( - 5, - 25 )