question about minimum value

Question

If y=x^2+8x+10 what is the minimum value of y

Answer 1

As can be seen there are various ways to solve this problem.

The equation as given...  y = x² + 8x + 10 ... is a parabola opening upward

Thus the minimum y value will be the y value at the point of the vertex.

 

Another way to find the vertex is to realize that with a quadratic equation

of the form ax² + bx + c = 0   the x value at the vertex is at x = -b/2a

 

To find the x coordinate of the vertex   x = -8/2 = -4

Plug -4 into the equation and solve for y

 

y = (-4)² + 8(-4) + 10 = 16 -32 + 10 = -6

 

The vertex is at (-4,-6) and the minimum y value is y = -6

Answer 2

To find the minimum value start by taking the derivative of the function. 

y = x² + 8x +10

y' = 2x + 8

 

Then set it equal to zero. The derivative tells us how fast a graph is changing at a certain point. When the derivative equals zero this tells you where the max and min of a function are.

2x + 8 = 0

x = -4

 

To find out whether -4 will give you a max or a min plug in a number greater than and less than -4. I'm going to us zero and negative infinity.

 

2(-∞) + 8 = a negative number. This means the graph is decreasing when x < -4

 

2(0) + 8 = a positive number. This means the graph is increasing when x > -4

 

So -4 is the value of x that gives us the minimum value of y

 

Now plug -4 in for x to y

y_min = (-4)² + 8(-4) + 10 = -6

Answer 3

y = x² + 8x + 10

 

Based on the first term of the function, we will have a positive parabola.  This type of graph will have a minimum.  To find the minimum, we must put this function in vertex form. 

 

y = a(x - h)² + k

 

where:

a is the coefficient of x² term

and the vertex is (h, k).  This coordinate will be the minimum.

 

 

y = (x² + 8x + 16) - 6

y = (x + 4)(x + 4) - 6

y = (x + 4)² - 6

 

h = -4

k = -6

 

The vertex is (-4, -6).

 

Therefore, the minimum value of y is -6.