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Let's break down the problem step by step:

### Problem 1: Find the equation of the parabola.
The given information is:
- The vertex is at \((-4, 15)\).
- The parabola passes through the point \((6, -35)\).
- The line of symmetry is parallel to the y-axis, which means the parabola has the form: 
  \[
  y = a(x - h)^2 + k
  \]
  where \((h, k)\) is the vertex of the parabola.

Given the vertex \((-4, 15)\), the equation becomes:
\[
y = a(x + 4)^2 + 15
\]

Now, we will use the point \((6, -35)\) to find the value of \(a\). Substituting \(x = 6\) and \(y = -35\) into the equation:
\[
-35 = a(6 + 4)^2 + 15
\]
\[
-35 = a(10)^2 + 15
\]
\[
-35 = 100a + 15
\]
\[
-35 - 15 = 100a
\]
\[
-50 = 100a
\]
\[
a = -\frac{1}{2}
\]

So, the equation of the parabola is:
\[
y = -\frac{1}{2}(x + 4)^2 + 15
\]

---

### Problem 2: When \(x = 16\), what is the value of \(y\)?
Substitute \(x = 16\) into the equation of the parabola:
\[
y = -\frac{1}{2}(16 + 4)^2 + 15
\]
\[
y = -\frac{1}{2}(20)^2 + 15
\]
\[
y = -\frac{1}{2}(400) + 15
\]
\[
y = -200 + 15
\]
\[
y = -185
\]

So, when \(x = 16\), \(y = -185\).

---

### Problem 3: Average rate of change between \(x = -4\) and \(x = 16\).
The average rate of change of a function between two points \((x_1, y_1)\) and \((x_2, y_2)\) is given by:
\[
\text{Average Rate of Change} = \frac{y_2 - y_1}{x_2 - x_1}
\]

We already know:
- At \(x = -4\), the vertex, \(y = 15\).
- At \(x = 16\), \(y = -185\).

Substitute these values:
\[
\text{Average Rate of Change} = \frac{-185 - 15}{16 - (-4)}
\]
\[
\text{Average Rate of Change} = \frac{-200}{16 + 4}
\]
\[
\text{Average Rate of Change} = \frac{-200}{20}
\]
\[
\text{Average Rate of Change} = -10
\]

So, the average rate of change between \(x = -4\) and \(x = 16\) is \(-10\).

---

Would you like more details or have any questions on this? Here are some follow-up questions:

1. What is the geometric significance of the value of \(a\) in the parabola equation?
2. How does the average rate of change relate to the slope between two points on a parabola?
3. Can you derive the vertex form of a parabola from the general quadratic equation?
4. What would happen if the line of symmetry were not parallel to the y-axis?
5. How would you calculate the focus of this parabola?

**Tip**: The value of \(a\) in the parabola's equation controls the "width" and the direction of the parabola's opening. A negative \(a\) means it opens downward.

A parabola that passes through the point (6, -35) has vertex (-4, 15). Its line of symmetry is parallel to the y-axis. Find equation of the parabola. When x = 16, what is the value of y? What is the average rate of change between x = -4 and x = 16?

Solve for the equation of a parabola passing through (6, -35) with vertex (-4, 15). Learn how to use the vertex form and calculate the average rate of change. Ideal for students in Grades 9-12.

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