SOLUTION: Given the functions: y= -x^2+5x-2 and y=x^2+5x-2 It is well known that the sign of the coefficient of the x^2 term determines whether the graph of the quadratic function wil

Algebra ->  Graphs -> SOLUTION: Given the functions: y= -x^2+5x-2 and y=x^2+5x-2 It is well known that the sign of the coefficient of the x^2 term determines whether the graph of the quadratic function wil      Log On


   

Question 1046618: Given the functions: y= -x^2+5x-2 and y=x^2+5x-2

It is well known that the sign of the coefficient of the x^2 term determines whether the graph of the quadratic function will have a "hill" or a "valley". On the basis of the present problem, which sign is associated with the hill? Supply an intuitive explanation for this.

Found 2 solutions by MathLover1, Theo:
Answer by MathLover1(20850) About Me  (Show Source):
You can put this solution on YOUR website!

if you have y=ax%5E2%2Bbx%2Bc
then its graph will either look like a ‘hill’ or a ‘valley’
· if ‘a’ is positive, it will be a valley
· if ‘a’ is negative, it will be a hill
so,
negative sign is associated with the hill
positive sign is associated with the valley
y=+-x%5E2%2B5x-2 -> y=+%28-1%29x%5E2%2B5x-2 ->y=+highlight%28-1%29x%5E2%2B5x-2->negative sign-> it will be a hill
and
y=x%5E2%2B5x-2->y=+%281%29x%5E2%2B5x-2->y=+highlight%281%29x%5E2%2B5x-2->positive sign-> it will be a valley


Answer by Theo(13342) About Me  (Show Source):
You can put this solution on YOUR website!
the negative coefficient of the x^2 term indicates that it will be a hill rather than a valley.

another way of describing it is that the quadratic equation opens down rather than up.

why is this?

x^2 is positive regardless if x is negative or positive.
therefore -x^2 is negative regardless if x is negative or positive.

the x^2 term is the most powerful term because, as x gets larger, the x^2 term gets progressively larger than any of the other terms in the equation.

it therefore dominates at both extremes of the graph.

here's what the graph of -x^2 + 5x - 2 looks like.



here's what the graph of x^2 + 5x - 2 looks like.



the quadratic equation will have a maximum or a minimum value at x = -b/2a.

this is derived from the standard form of the quadratic equation which is ax^2 + bx + c = 0

when the equation is -x^2 + 5x - 2, the maximum value of the equation is when x = -5/-2 = 2.5.

the value of y would be -(2.5)^2 + 5*(2.5) - 2 = 4.25

when the equation is x^2 + 5x - 2, the minimum value of the equation is when x = -5/2 = -2.5.

the value of y would be (-2.5)^2 -5*2.5 - 2 = -8.25

you can see that from the graphs.