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

To find the possible value of \( a + b + c \) for the parabola with vertex \((9, -14)\) that intersects the x-axis at two points, we can start by using the vertex form of a parabola, which is given by:

\[
y = a(x - h)^2 + k
\]

where \((h, k)\) is the vertex. Here, \((h, k) = (9, -14)\), so the equation becomes:

\[
y = a(x - 9)^2 - 14
\]

To find the x-intercepts (where \( y = 0 \)), we set the equation to zero:

\[
0 = a(x - 9)^2 - 14
\]

This simplifies to:

\[
a(x - 9)^2 = 14
\]

Thus, we have:

\[
(x - 9)^2 = \frac{14}{a}
\]

For the parabola to intersect the x-axis at two points, \( \frac{14}{a} \) must be positive, implying that \( a \) must be positive. Now, taking the square root gives:

\[
x - 9 = \pm \sqrt{\frac{14}{a}}
\]

Thus, the x-intercepts are:

\[
x = 9 \pm \sqrt{\frac{14}{a}}
\]

Now, we can convert the vertex form to standard form \( y = ax^2 + bx + c \). We expand the vertex form:

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

This expands to:

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

From this, we identify:

- \( b = -18a \)
- \( c = 81a - 14 \)

Now, we can find \( a + b + c \):

\[
a + b + c = a - 18a + (81a - 14) = (a - 18a + 81a - 14) = 64a - 14
\]

Next, we can analyze the possible values of \( 64a - 14 \):

1. For \( a = \frac{1}{4} \):
   \[
   64 \times \frac{1}{4} - 14 = 16 - 14 = 2
   \]

2. For \( a = 1 \):
   \[
   64 \times 1 - 14 = 64 - 14 = 50
   \]

3. For \( a = \frac{1}{2} \):
   \[
   64 \times \frac{1}{2} - 14 = 32 - 14 = 18
   \]

Since we need to determine values of \( a + b + c \) that could equal one of the options provided:

- **A)** -23
- **B)** -19
- **C)** -14
- **D)** -12

To find \( a + b + c \) equal to these options, we set \( 64a - 14 = k \) (where \( k \) is one of the options) and solve for \( a \):

1. For \( -23 \):
   \[
   64a - 14 = -23 \implies 64a = -9 \implies a = -\frac{9}{64} \quad \text{(not possible, as } a > 0\text{)}
   \]

2. For \( -19 \):
   \[
   64a - 14 = -19 \implies 64a = -5 \implies a = -\frac{5}{64} \quad \text{(not possible)}
   \]

3. For \( -14 \):
   \[
   64a - 14 = -14 \implies 64a = 0 \implies a = 0 \quad \text{(not valid)}
   \]

4. For \( -12 \):
   \[
   64a - 14 = -12 \implies 64a = 2 \implies a = \frac{1}{32} \quad \text{(valid, } a > 0\text{)}
   \]

Thus, the only valid value is:

\[
\boxed{-12}
\]

Would you like any further details or explanations on this topic? Here are some questions to consider:

1. How does the vertex affect the graph of a parabola?
2. What is the significance of the x-intercepts in a parabola?
3. Can you derive the equation of a parabola given different sets of points?
4. How do you determine whether a parabola opens upwards or downwards?
5. What are the applications of parabolas in real life? 

**Tip:** Always check the sign of \( a \) to determine the direction in which the parabola opens.

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?

Explore how to determine the possible values of a + b + c for a parabola with vertex (9, -14) that intersects the x-axis at two points. This detailed analysis includes the vertex form of the parabola and the corresponding standard form. Suitable for Grades 9-12 students.

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