Part A:
The axis of symmetry of a quadratic function is a vertical line that divides the parabola into two symmetric halves. It passes through the vertex of the quadratic function. On the other hand, the quadratic formula is a mathematical equation used to find the roots of a quadratic function. It states that the roots of a quadratic function can be calculated using the formula:
x = (-b ± √(b^2 - 4ac))/(2a)
Now, let's see how these two concepts are related. The axis of symmetry of a quadratic function can be represented by the equation x = h, where h is the x-coordinate of the vertex of the parabola. In the general quadratic function form, f(x) = ax^2 + bx + c, with a, b, and c being constants, the x-coordinate of the vertex is given by h = -b/(2a).
By comparing the equation x = h with the quadratic formula, we can observe that the x-coordinate of the vertex (-b/(2a)) is in fact the value of x that yields the maximum or minimum point of the parabola. Therefore, the axis of symmetry and the quadratic formula provide information about the x-coordinate of the vertex, allowing us to find the roots of a quadratic function.
Part B:
The quadratic formula, which is x = (-b ± √(b^2 - 4ac))/(2a), provides a way to determine the roots of a quadratic function. The discriminant, which is the value inside the square root, b^2 - 4ac, plays a crucial role in determining the number of roots.
If the discriminant is positive (b^2 - 4ac > 0), then the quadratic function has two distinct real roots. Each root corresponds to one of the ± signs in the quadratic formula.
If the discriminant is zero (b^2 - 4ac = 0), then the quadratic function has exactly one real root. In this case, the ± sign in the quadratic formula becomes irrelevant, as both the added and subtracted versions yield the same value.
If the discriminant is negative (b^2 - 4ac < 0), then the quadratic function has no real roots. This is because the square root of a negative number is not a real number.
By examining the value of the discriminant, we can determine whether the quadratic function has zero, one, or two real roots without actually finding the roots themselves. This information helps us understand the behavior and properties of the quadratic equation.