Hiya! I have a function here:

Question

y = x^2 - 4x - 32

I have some questions about it.

1. I think this parabola opens up. Am I correct? Can you explain how you know weather it opens up or down?

2. Can you tell me what the axis of symmetry and the vertex of this parabola are?

3. Can you tell me the minimum?

4. Could you show me how to find the x-intercepts by factoring?

5. How does the value of the discriminant
(what I know to be b^2 - 4ac) support the conclusion in the previous question?

6. How does the axis of symmetry relate to the x-intercepts?

I know there is quite a bit here. I really appreciate your help! :D

Answer 1

first, it's "whether" not "weather".
Look at the coefficient of x^2. Since x^2 is always positive, the parabola grows upward if the coefficient is positive, and downward if it is negative.

The easy way to find the vertex and axis is to complete the square:

y = x^2 - 4x - 32
= x^2-4x+4 - 28
= (x-2)^2 - 28
So the vertex is at (2,-28)
the axis is x=2

Naturally, the minimum is at the vertex: -28

Check the factors of 32. We want two of them which differ by 4.
(x-8)(x+4)

Not sure how the discriminant relates to #4. If it's positive, you may be able to factor it; if not, there are no real roots, so no way to factor it.

The axis of symmetry is midway between the roots (duh - the roots are symmetric about the axis!), at x=2

Answer 2

Thank you. Also thanks for correcting the "whether" VS. "weather" mistake. I always have trouble with that. :)