Question

The functions f, g and h if graphs are drawn in the coordinate below the regulation of the form p (x) = ax ^ 2 + bx + c

Determine the sign of a, b​​, c and d (where d is the discriminant). Give reasons for your answer.

Answer 1

This is the graph

Answer 2

@whpalmer4, is this something you can help me with?

Answer 3

or you, @AravindG ?

Answer 4

For all cup shaped quadratics a is positive. When it is inverted sign of a is negative.

Answer 5

Okay, that makes sense... What about b, c and d? @AravindG

Answer 6

Let's deal with d first. When graph cuts the x axis at two points we say 2 roots occur and hence d>0. When it cuts in only 1 point d=0 and when it doesnt cut any point d<0.

Answer 7

Keeping in mind that d=b^2-4ac and the sign of a, you can predict sign of b,c for the functions given.

Answer 8

So for an example, graph g: a is negative and d=0?

Answer 9

No, wait d<0 for d, because it doesn't cut the x axis, right? @AravindG

Answer 10

^^Exactly. You get a medal for that :)

Answer 11

Haha, thanks! it's the same for f too? It doesn't look like it's going the x axis, and a should be positive, right? And graph h, a is positive with d>0.. right? @AravindG

How do you find b then? isn't c, where the graph cuts the y-axis?

Answer 12

For f it definitely touches x axis at one point. So it is not same cases as g. Yes a is positive for f.

Answer 13

You are right about h by the way :)

ok I will show you an example of finding sign of b and c taking the function f:

d=b^2-4ac
d=0 as cuts only in 1 point.
This means b^2=4ac
Now we know a is positive and b^2 is always positive. Hence c is also positive.

Answer 14

I'm awarding a medal for the same reason, but I must say that @AravindG 's write-up here is very fine, and I wish I could award two medals!

Answer 15

(If I may add another method).

Let's find the sign of \(b\) in \(h(x)\).

Both roots positive, so the sum of roots is positive.

So \(-b/a\) is positive. You know the sign of \(a\) (positive), so \(b\) must be negative.

Answer 16

Similarly the sum of roots in f is negative...

Answer 17

@AravindG, I'm not fully with you... How did you get from d=b^2-4ac to b^2=4ac...

Answer 18

@ParthKohli, so what you're saying is that if a is positive b should be negative? always?

Answer 19

Thank you @whpalmer4 :)

Recall that for a graph that cuts only at one point d=0 Put it in equation for d ie d=b^2-4ac-->b^2=4ac

Answer 20

\[-b/a > 0\]-b/a is the sum of roots; a is positive because the graph points up so we can just remove the positive denom from the inequality\[-b> 0\]\[b < 0\]this is for h(x).

Answer 21

do you see how this was done?

Answer 22

A little confused.. @ParthKohli

Answer 23

hmm, do you know the formula sum of roots and product of roots of a quadratic equation?

Answer 24

sum of roots is -b/a
product of roots is c/a

^ that

Answer 25

@AravindG so if i do that, i'll know a,b,c and d?

Answer 26

No, but I guess I know now? @ParthKohli -

Answer 27

but this is only an alternative. so you should follow what you were taught.

Answer 28

@ParthKohli, I take online lessons - and there isn't really much of an method :/

Answer 29

You get signs of a,c,d.

For sign of b I just look at the axis of symmetry. If it is having a negative x value b becomes positive and vice versa.

Answer 30

^For a<0 Do the opposite of what is said above.

Answer 31

@AravindG so if a was positive, b would be negative? But how do I explain how i know this?

Answer 32

no sweat :)

it's a very useful result though. here's how we derive it:

let \(p\) and \(q\) be the roots/zeroes of a quadratic \(x^2 + bx + c= a\left(x^2 + b/a ~x ~+ c/a\right)\). by the factor theorem, the polynomial can be written as \(a(x-p)(x-q) = a(x^2 - (p+q)x + pq)\).

so \(a(x^2 + b/a ~x + c/a) =a(x^2 - (p+q)x + pq)\).

look at the coefficients and see how\[p+q = b/a\\pq = c/a\]you can use this now to show what aravindg said.

Answer 33

you can also use this to find c!

Answer 34

but c is the value of y when x = 0, so let's keep it simple lol

Answer 35

Using the axis of symmetry concept!

Always draw a vertical line that cuts the graph into two. Look at the point where it touches x axis. The sign of this value is important. For example For graph h it is negative. Once you know the sign its easy

x intercept of axis of symmetry=-b/2a
Put in the sign and we can do all comparisons.

For h,

-ve=-b/2(+ve)
Hence b=-ve.

Answer 36

@AravindG, okay! Thanks!
I'll see if i can write into words in my language, lol.
@Parthkohli Thanks a bunch too! I'll tag you if i need more help, lol.

Answer 37

yw :) Indeed these notes can prove very useful in the end exam. Good luck!

Answer 38

@AravindG, yes, indeed! Thanks!