Question

How do you find two quadratic function one that opens up and one that opens downward whose graphs have intercepts (4,0), (8,0)?

Answer 1

`y=x^2-12x+32larr" upwards"`
`y=-x^2+12x-32larr" downwards"`

Explanation:

There are an infinite number of quadratic functions possible.

Since we know the x-intercepts (4 ,0) and (8 ,0)

`rArr"roots are " x=4" and " x=8`

`"and the factors are "(x-4),(x-8)`

Hence the general equation is.

`y=a(x-4)(x-8)`

• If a > 0 , then graph opens up

• If a < 0 , then graph opens down

To find the particular function we require to know another point

For this example, lets choose a = 1 and a = - 1

`a=1toy=x^2-12x+32larr" distributing brackets"`

`a=-1toy=-(x^2-12x+32)=-x^2+12x-32`
graph{(y-x^2+12x-32)(y+x^2-12x+32)=0 [-10, 10, -5, 5]}