Question

if the curve : Y = X (a - X), which of the following statements could be right? a) the curve intersects the X-axis at the two points (0,0) (a,0) b) the curve vertex is (a/2 , a^2/4) c) the axis of symmetry of the curve is: X = a here’s the catch: it’s either that two of these choices are correct or all of them are , my teacher also said it has something to do with the rule : X = -b/2a , Y= f(-b/2a)

Answer 1

(a) and (b) are true. (c) is false

Explanation:

For (a) we need to find the x intercepts which happens when y=0

x(a-x) = 0

either x = 0 or a-x = 0 which means x=a

so the x intercepts are (0,0) and (a,0) making (a) true

For (b) we need to first find the x coordinate of the vertex given by:

-b/2a. By expanding the bracket turn your equation into y = -
`x^(2)`+ax. Here your b = a and your a = -1

The result is -a/(2×-1) = a/2 (by the way this is what makes part (c) false since the axis of symmetry is equivalent to the x coordinate of the vertex)

Substitute a/2 into the original equation a/2(a-a/2) =
`a^(2)`/4 so the vertex is at (a/2 ,
`a^(2)`/4)

Part (c) is false as mentioned earlier.