Question

In the xy-plane, a parabola has vertex (9, −14) and

intersects the x-axis at two points. If the equation of
the parabola is written in the form y = ax2 + bx + c,
where a, b, and c are constants, which of the
following could be the value of a b + + c ?

Answer 1

Here is your answer.

Step-by-step explanation:

Since the parabola intersects the x-axis at two points, it means that the parabola opens upwards. The vertex of the parabola is (9, -14), which is the lowest point on the curve.

The general form of a parabolic equation that opens upwards is: y = a(x - h)^2 + k, where (h, k) is the vertex.

Given the vertex (9, -14), the equation of the parabola can be written as:

y = a(x - 9)^2 - 14

Now, we can expand the squared term:

y = a(x^2 - 18x + 81) - 14

y = ax^2 - 18ax + (81a - 14)

Comparing this with the form y = ax^2 + bx + c:

We have: b = -18a and c = 81a - 14

Now we need to find the value of "a * b + c":

a * b + c = a * (-18a) + (81a - 14)

a * b + c = -18a^2 + 81a - 14

We don't have a specific value for "a" to work with, so we can't calculate the exact numerical value of the expression -18a^2 + 81a - 14. However, we can still determine some properties of this expression.

The coefficient of the quadratic term is negative (-18a^2), which means that the parabola opens downwards when "a" is positive and opens upwards when "a" is negative. Given that the parabola opens upwards, "a" should be negative.

Also, the expression is a quadratic, and its coefficient is negative. This implies that it has a maximum point, and the vertex of the parabola is at its maximum. This aligns with the information given that the vertex of the parabola is the lowest point, which is consistent with the parabola opening upwards.

In conclusion, we can't determine a specific value for the expression a * b + c without knowing the value of "a." However, we do know that "a" should be negative based on the given information.

Answer 2

The equation of the parabola is y = a(x-9)^2 - 14. The discriminant of the equation must be greater than zero. The correct value of a*b+c is A.

Step-by-step explanation:

The equation of a parabola in the form y = ax^2 + bx + c can be determined using the given information. The vertex form of a parabola is given by y = a(x-h)^2 + k where (h,k) represents the vertex. So, we can deduce that the equation of the parabola is y = a(x-9)^2 - 14. Since the parabola intersects the x-axis at two points, the discriminant of the equation b^2 - 4ac must be greater than zero. This gives us the inequality 4a(14+14) > 0. Evaluating this expression, we find that a > 0, which means that option A is the correct answer.