Hiya! I have a function here:

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

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

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

Hello! I'd be happy to help you with your questions about the function y = x^2 - 4x - 32. Let's go through each question step by step:

1. To determine whether a parabola opens up or down, we can look at the coefficient of the x^2 term. In this case, the coefficient of x^2 is positive (1), which means that the parabola opens upwards. When the coefficient of x^2 is positive, the parabola opens upwards, and when it's negative, the parabola opens downwards.

2. To find the axis of symmetry and the vertex of a parabola in the form y = ax^2 + bx + c, we can use the formula: x = -b / (2a) to find the x-coordinate of the vertex. In our case, a = 1 and b = -4. Plugging these values into the formula, we have: x = -(-4) / (2*1) = 2. So the axis of symmetry is x = 2. To find the y-coordinate of the vertex, we substitute this x-value into the equation to get y = (2)^2 - 4(2) - 32 = -8. Therefore, the vertex is (2, -8).

3. To find the minimum of the parabola, we look at the y-coordinate of the vertex. In this case, the y-coordinate of the vertex is -8. Therefore, the minimum value of the function is -8.

4. To find the x-intercepts of the parabola by factoring, we set y = 0 in the equation and solve for x. So we have x^2 - 4x - 32 = 0. This equation can be factored as (x - 8)(x + 4) = 0. Setting each factor equal to zero gives us x - 8 = 0 and x + 4 = 0. Solving these equations, we find x = 8 and x = -4. So the x-intercepts of the parabola are x = 8 and x = -4.

5. The value of the discriminant (b^2 - 4ac) helps us determine the nature of the roots of a quadratic equation. In this case, the equation is x^2 - 4x - 32 = 0. Comparing with the general quadratic equation ax^2 + bx + c = 0, we can see that a = 1, b = -4, and c = -32. The discriminant is given by b^2 - 4ac, which in this case is (-4)^2 - 4(1)(-32) = 16 + 128 = 144. Since the discriminant is positive (greater than 0), there are two distinct x-intercepts, supporting the fact that the parabola opens upwards and has a minimum.

6. The axis of symmetry is a vertical line that passes through the vertex of the parabola. In this case, the axis of symmetry is x = 2. The x-intercepts occur where the parabola intersects the x-axis. Since the axis of symmetry is the line of symmetry for the parabola, it is equidistant from each x-intercept. In other words, the x-value of the axis of symmetry is the average of the x-values of the x-intercepts. In this case, the x-intercepts are 8 and -4, so their average is (8 + (-4)) / 2 = 4 / 2 = 2, which matches the x-value of the axis of symmetry.

I hope that helps! Let me know if you have any further questions.