Question

Which of the following statements are always true?

I. For a parabola written in standard form (y=ax^2+bx+c), if a > 0, the vertex represents the absolute minimum point on the parabola.
II. Every parabola has one or more real zeros.
III. For a parabola with two real zeros, you can find the coordinates of the vertex by finding the midpoint between the roots, and evaluating the function at this x-value.
A) I and II
B) II and III
C) I and III
D) All of the above

Answer 1

The correct answer to the question is C) I and III, as statements I and III about parabolas are always true, but statement II is not since parabolas can have complex roots with no real zeros.

Step-by-step explanation:

You asked which statements are always true regarding parabolas, and here are the explanations to determine the correct answer from the options provided:

1. For a parabola written in standard form (y=ax^2+bx+c), if a > 0, the vertex does indeed represent the absolute minimum point on the parabola. This is because a positive 'a' value opens the parabola upwards, making the vertex the lowest point.
2. It's not true that every parabola has one or more real zeros. Some parabolas do not intersect the x-axis at all, especially when the discriminant (b^2-4ac) is negative, resulting in complex roots.
3. For a parabola with two real zeros, you can find the coordinates of the vertex by finding the midpoint between the roots, which gives you the x-value. Then, evaluating the function at this x-value will give you the y-coordinate of the vertex.

Considering these points, the correct answer is C) I and III.