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We are given the quadratic function \( f(x) = ax^2 + bx + c \), and it passes through the points \( (0, -8) \), \( (3, 10) \), and \( (6, 34) \). The question asks to find the value of the function when \( x = -3 \).

### Step 1: Set up the system of equations
Using the points provided, we can set up three equations for \( a \), \( b \), and \( c \).

1. At \( x = 0 \), \( f(0) = -8 \):
   \[
   a(0)^2 + b(0) + c = -8 \implies c = -8
   \]

2. At \( x = 3 \), \( f(3) = 10 \):
   \[
   a(3)^2 + b(3) + c = 10 \implies 9a + 3b - 8 = 10 \implies 9a + 3b = 18
   \]
   Simplifying this equation:
   \[
   3a + b = 6 \quad \text{(Equation 1)}
   \]

3. At \( x = 6 \), \( f(6) = 34 \):
   \[
   a(6)^2 + b(6) + c = 34 \implies 36a + 6b - 8 = 34 \implies 36a + 6b = 42
   \]
   Simplifying this equation:
   \[
   6a + b = 7 \quad \text{(Equation 2)}
   \]

### Step 2: Solve the system of equations
Now, subtract Equation 1 from Equation 2:
\[
(6a + b) - (3a + b) = 7 - 6
\]
\[
3a = 1 \implies a = \frac{1}{3}
\]

Substitute \( a = \frac{1}{3} \) into Equation 1:
\[
3\left(\frac{1}{3}\right) + b = 6 \implies 1 + b = 6 \implies b = 5
\]

So, we now have:
- \( a = \frac{1}{3} \)
- \( b = 5 \)
- \( c = -8 \)

### Step 3: Find \( f(-3) \)
Now that we know \( a \), \( b \), and \( c \), we can find the value of the function when \( x = -3 \):
\[
f(-3) = \frac{1}{3}(-3)^2 + 5(-3) - 8
\]
\[
f(-3) = \frac{1}{3}(9) - 15 - 8 = 3 - 15 - 8 = -20
\]

Thus, the value of the function when \( x = -3 \) is \( \boxed{-20} \).

Would you like further details or explanations? Here are five related questions:

1. How do you solve quadratic systems with three points?
2. Can you explain how to eliminate variables in a system of equations?
3. How does the quadratic formula relate to finding the coefficients of a parabola?
4. What are other methods for solving systems of equations?
5. Can the vertex form of a quadratic equation help in solving this?

**Tip:** Always start by solving for constants like \( c \) first when the equation allows for an easy substitution.

The graph of a quadratic function having the form f(x) = ax^2 + bx + c passes through the points (0, -8), (3, 10), and (6, 34). What is the value of the function when x = -3?

Learn how to solve a quadratic function passing through specific points (0, -8), (3, 10), and (6, 34) and find the value of the function at x = -3. Detailed step-by-step process for solving systems of equations to determine a, b, and c in f(x) = ax^2 + bx + c. Suitable for students in Grades 9-11.

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