What is the difference between Maximum and Minimum in Quadratic Equations?

Question

What are the differences from finding one versus the other?

Answer 1

A quadratic equation is in the shape of a parabola.

if you look at the a(x)^2 + bx + c form of the equation, you can learn a lot.

If the coefficient of the x^2 term, the "a," is positive, the parabola opens UP like a "U" and thus has a bottom tip, or MINIMUM. The x value of that minimum is -b/2a.

On the other hand, if "a" is negative, the parabola opens DOWN, like an upside down U, and thus has a top tip, or MAXIMUM. Again, the x value of that maximum is -b/2a.

(to find the y value, simply substitute the x value you found above into the original equation)

A kind of neat sample of graphing software which will graph an equation and show you the maximum OR minimum is shown below. It shows a sample equation which opens UP and thus has a MINIMUM. (they call it the vertex in this explanation)

Answer 2

they are almost they same. you take the first derivative of y with respect to x and equate y' to 0 then solve for x.

for example,
y = x^2 -2x +1
y' = 2x - 2
0 = 2x-2
x = 1

But in order for you to check whether that value of x is a max or a min, you have to take the second derivative of the original equation.

y = x^2 - 2x +1
y' = 2x - 2
y" = 2

if y">0 then the x value you got is the x-coodinate of the min value of y.
if y"<0 then the x value you got is the x-coordinate of the max value of y.
if y"=0 then the x value is the x coordinate of the point of inflection of the graph.

since we have, y" = 2 >0 then
then x=1 is the x -coordinate of the min value of y which is,


y = x^2 -2x +1
y = (1)^2 -2(1) +1
y = 1 - 2 +1
y = 0

the minimum point is at (1,0) and the curve opens upward.

Answer 3

Basically, the maximum value is the highest number that can solve the equation. The minimum value is the lowest number that can solve the equation.

Answer 4

Max opens down (upside down "U")
Min opens up (right side up "U")