Question
Quadratic function written in standard form where a, b, and c are constants such that a is not zero.
f(x)= ax^2+bx+c
Using calculus find the vertex of the parabola formed by this quadratic function. Determine under what conditions this vertex is a maximum or minumum (using Calculus techniques). Show work using derivatives to justify our conclusions.
THis is what I did so far, but can't figure out the rest.
f'(x)= 2ax + b
0=2ax+b
-b = 2ax
-b/2a = x
Please help and show steps so that I can understand.
f(x)= ax^2+bx+c
Using calculus find the vertex of the parabola formed by this quadratic function. Determine under what conditions this vertex is a maximum or minumum (using Calculus techniques). Show work using derivatives to justify our conclusions.
THis is what I did so far, but can't figure out the rest.
f'(x)= 2ax + b
0=2ax+b
-b = 2ax
-b/2a = x
Please help and show steps so that I can understand.
Answers
That is correct. The minimum ofr maximum (vertex) is at x = -b/2a
You already know that from the quadratic equation. x = -b/2a +/- (b/2a)sqrt (b^2-4ac)
Now
if x = -b/2a
what is y?
y = a (b^2/4a^2) + b(-b/2a) + c
= b^2/4a -b^2/2a + c
= -b^2/4a+c
for max or min
f" = 2 a
if a is +, that is a minimum
if a is -, that is a maximum
You already know that from the quadratic equation. x = -b/2a +/- (b/2a)sqrt (b^2-4ac)
Now
if x = -b/2a
what is y?
y = a (b^2/4a^2) + b(-b/2a) + c
= b^2/4a -b^2/2a + c
= -b^2/4a+c
for max or min
f" = 2 a
if a is +, that is a minimum
if a is -, that is a maximum
Related Questions
1. How do we determine te quadratic function given its table of values?
2. How can we use the gen...
The solutions of a quadratic equation are −4 and 9. What is the standard form of a related quadratic...
The table represents a quadratic function f(x).
x f(x)
−10 24
−9 17
−8 12
−7 9
−6 8
−5 9...
Determine the value of the constant term of the quadratic function in standard form
The value of c...