https://file.aiquickdraw.com/mathbot/file/1727111656049g7ljpiev.jpg

We are asked to find the value of \( a + b + c \) for a parabola with a vertex at \( (9, -14) \), written in the form \( y = ax^2 + bx + c \), and which intersects the x-axis at two points.

### Step 1: Vertex Form of a Parabola
The vertex form of a parabola is:
\[
y = a(x - h)^2 + k
\]
where \( (h, k) \) is the vertex. Given that the vertex is \( (9, -14) \), we can write the equation as:
\[
y = a(x - 9)^2 - 14
\]

### Step 2: Convert to Standard Form
To match the given standard form \( y = ax^2 + bx + c \), expand the vertex form:
\[
y = a(x^2 - 18x + 81) - 14
\]
\[
y = ax^2 - 18ax + 81a - 14
\]
So, in standard form:
\[
y = ax^2 + (-18a)x + (81a - 14)
\]

Thus, comparing with \( y = ax^2 + bx + c \), we have:
- \( b = -18a \)
- \( c = 81a - 14 \)

### Step 3: Find \( a + b + c \)
Now, calculate \( a + b + c \):
\[
a + b + c = a + (-18a) + (81a - 14) = a - 18a + 81a - 14 = 64a - 14
\]

### Step 4: Use the fact that the parabola intersects the x-axis at two points
Since the parabola intersects the x-axis at two points, the discriminant of the quadratic equation must be positive. This implies \( b^2 - 4ac > 0 \). However, this information is generally to confirm that there are real roots, not for direct calculation. The question asks us to find possible values for \( a + b + c \).

### Step 5: Check Options
We are given four options for \( a + b + c \):
- A) -23
- B) -19
- C) -14
- D) -12

Substitute these into \( 64a - 14 = \text{option value} \):

\[
64a - 14 = -23 \quad \Rightarrow \quad 64a = -9 \quad \Rightarrow \quad a = -\frac{9}{64}
\]
\[
64a - 14 = -19 \quad \Rightarrow \quad 64a = -5 \quad \Rightarrow \quad a = -\frac{5}{64}
\]
\[
64a - 14 = -14 \quad \Rightarrow \quad 64a = 0 \quad \Rightarrow \quad a = 0
\]
\[
64a - 14 = -12 \quad \Rightarrow \quad 64a = 2 \quad \Rightarrow \quad a = \frac{2}{64} = \frac{1}{32}
\]

For real-world situations, \( a = 0 \) would not result in a parabola, so the valid solution is \( a + b + c = -23 \).

Thus, the answer is **A) -23**.

Would you like any further details or clarification?

Here are some related questions to deepen understanding:
1. How does the discriminant determine the number of x-axis intersections for a parabola?
2. What is the significance of the coefficient \( a \) in the equation of a parabola?
3. How would the equation change if the parabola did not intersect the x-axis?
4. Can we deduce the exact x-intercepts from this form of the equation?
5. How does the vertex form help in determining the vertex directly from an equation?

**Tip:** Always check if a parabola intersects the x-axis by analyzing the discriminant \( b^2 - 4ac \). If positive, it has two real solutions!

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 = ax^2 + bx + c, where a, b, and c are constants, which of the following could be the value of a + b + c?
A) -23
B) -19
C) -14
D) -12

Explore how to find the value of a + b + c for a parabola with vertex (9, -14) intersecting the x-axis at two points. Learn about the vertex form, standard form, and the use of the discriminant for quadratic equations. Suitable for Grades 10-12.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation