maths question - quadratic graphs

can someone help me with this question please??

You have three points the function and its derivative passes through, so you have 3 linear equations in three unknowns (a,b,c) to solve.

Or, you can start by reasoning that if the function has one root at x = 4 and its minimum is at x = 5, then, by symmetry, the other root must be at x = ....

(6,0)? now i can sub -4 and -6 and the solve it right?

Or write the quadratic down in terms of its factors (roots) and expand (with a multiplier on the front to get the value of the minimum right).

can someone help me with this question please??

If it has a minimum point of (5,-3) and if it passes through (4,0) then this means that it crosses the x axis at 4.
We can work this out by completing the square backwards, and then by using the factor theorem!

y = k(x-5)^2 - 3

y = k(x-5)(x-5) - 3

y = k(x^2-10x+25) - 3

y = kx^2 - 10kx + 25k - 3

but i only have 4 what's the other root?

If it has a minimum point of (5,-3) and if it passes through (4,0) then this means that it crosses the x axis at 4.
We can work this out by completing the square backwards, and then by using the factor theorem!

idek what the factor theorem is lol

The roots are symmetric about the minimum so the other one is 6.
x = 4, 6
so the factors are
(x-4), (x-6)
If a = 1, that would give the quadratic when you expand them? So
a(x-4)(x-6)
is the quadratic. Use the minimum point to determine "a".



Hints not solutions

(6,0)? now i can sub -4 and -6 and the solve it right?

Oh lol... you don't really need to know what it means.

Basically, if you know one root and have an equation (as we created with the value k), you substitute the root you know into it and make it equal to 0. When x is 4, y is 0. If we know this, we can work out what k is. After that, we just substitute k back into the equation.

Yea thats what i was trying to say b4


thank you but im gonna stick to the easier version 😭 i still dont get your method. am i supposed to know the factor theorem?

It’s super easy dw. If you haven’t learnt it then you either will or don’t need to. Let me go through it again in a more easy way:

We expanded and got this as our equation:
y = kx^2 - 10kx + 25k - 3

We know that when x = 4, y = 0 (4,0) so we substitute 4 into x, and 0 into y:

0= 16k - 40k + 25k - 3

0 = k - 3

Therefore k = 3

And after we know what k is, we substitute it back into the equation we made at the beginning

oh i get it
do u know where i can use this theorem? cuz i dont i will remember it i need to do some practice

You use it in mostly questions like these. There might be a few harder questions but if you’ve got to find a quadratic expression, then it’ll come in useful.

Just remember it for questions like these that need you to work out the equation of a graph. I don’t really know how else you could use it