Answer:
all real values of x where 1 < x < 4
If there is anything you don't understand let me know. Also, since the graph is supposedly shown you may even be able to get the information just by looking at the graph.
Step-by-step explanation:
A graph goes from being negative to positive (or the other way around) by passing through the x axis. Or in other words when f(x) = 0. So the trick is to find all 0s and then test if it is poitive or negative before and after the 0.
f(x) = (x+2)(x-4) is in factored form, so it gives the 0s. in factored form the 0s are the negatives of the numbers int he parenthesis. Or in other words (x+a)(x+b) would have 0s at -a and -b. Notie it is x+a but the 0 is at -a. so for (x+2)(x-4) the zeroes would be -2 and 4.
Then checking, we can say it is positive before -2, negative after -2 up until 4, then it is positive again.
For increasing and decreasing for anything other than a quadratic and linear function you need calculus. Since this is a quadratic though you don't need it.
With quadratics you need to know they have two parts. An increasing part and decreasing part. If you draw the graph of x^2 you can see it decreases until (0,0) then it starts increasing. -x^2 is the opposite. it won't always have the same changing point, but you can always tell if it increases first or decreases first. You look at if the coefficient next to x^2 is positive of negative.
In (x+2)(x-4) We have to expand first. so it becomes x^2 - 2x - 8. x^2 has 1 as its coefficent so we know it decreases first then increases. Also looking at (x+2)(x-4) you could say since the two xs have the same sign we also know it would be that. is one was positive and one negative, like (2x+1)(-3x+1) we would be able to say it increases first then decreases.
To know when it switches, since it is in factored form you find the point in the middle of the two zeroes, and that is where it switches. Since the zeroes are -2 and 4 we can say the middle (called the vertex) is at x=1 since it is 3 away from both 0s. So it decreases up until 1 then increases.
The different forms of quadratics have different easy methods of finding the vertex.
We could also actually use this information to say, since it is decreasing first we know it is positive before the first 0, then negative until the second 0, then positive again. Still would have to find the 0s though.
So now we know it is positive and decreasing until x=-2, negative and decreasing until 1, negative and increasing until 4, then increasing after that.
Since you want negative and increasing, that is 1 < x < 4
