When we are studying a function and trying to figure how it looks like we usually study the sign of this function, putting f(x) > 0. And then we will have a region of the graph where it's positive and region where it's negative. I've always though that this was supposed to show me where the function was going upwards and where it was going downwards. But I've done some exercises and they don't correspond to this.
So...how do I know when a function is going up, when it's going down and specially when it's just flat???????
10 points for the one who enlightens me!
So...how do I know when a function is going up, when it's going down and specially when it's just flat???????
10 points for the one who enlightens me!
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A function is decreasing when its slope is negative. For example, graph y=-x and notice the direction in which the graph goes, because for all of this function slope is negative. Graph y=x and take a look at the direction of the positive slope for the entire line. Using more limit reasoning, If a function tends toward infinity as x goes to infinity, the function is increasing.
If you have any calculus background, take the derivative of the function. The derivative gives the instantaneous slope at any point:
i. Using the rule, the derivative of ax^b is bax^(b-1):
consider the polynomial
y=3x^2 + 2x + 1
Whose derivative is y'=6x + 2
Where y' is positive, the function is increasing, where its negative the function is decreasing.
ii. When the function is "flat" its slope is zero.
You may know the formula for the x coordinate of a parabolas vertex is x=-b/2a? But why?
A parabola is in the form y=ax^2 + bx + c
The derivative is y'=2ax + b
You want to solve for where slope is zero? set y to zero.
0=2ax + b
-b = 2ax
x=-b/2a
This finds maximum / minimum of a parabola.
In general. If y is increasing as x is increasing, the function is increasing. If y is decreasing as x is increasing, the function is decreasing. I hope this helps.
If you have any calculus background, take the derivative of the function. The derivative gives the instantaneous slope at any point:
i. Using the rule, the derivative of ax^b is bax^(b-1):
consider the polynomial
y=3x^2 + 2x + 1
Whose derivative is y'=6x + 2
Where y' is positive, the function is increasing, where its negative the function is decreasing.
ii. When the function is "flat" its slope is zero.
You may know the formula for the x coordinate of a parabolas vertex is x=-b/2a? But why?
A parabola is in the form y=ax^2 + bx + c
The derivative is y'=2ax + b
You want to solve for where slope is zero? set y to zero.
0=2ax + b
-b = 2ax
x=-b/2a
This finds maximum / minimum of a parabola.
In general. If y is increasing as x is increasing, the function is increasing. If y is decreasing as x is increasing, the function is decreasing. I hope this helps.