The problem involves finding the values of $a$ and $b$ for a quadratic equation given the minimum point $(8, -3)$ and the equation in the form $y = (x + a)^2 + b$.

Step 1: Relationship of the Minimum Point

The vertex (minimum point for a parabola that opens upward) of a quadratic function in the form $y = (x + a)^2 + b$ occurs at: $x = -a \quad \text{and} \quad y = b.$

Here, the given minimum point is $(8, -3)$. Thus: $-a = 8 \quad \Rightarrow \quad a = -8,$ and: $b = -3.$

Step 2: Final Equation

Substituting $a = -8$ and $b = -3$ into the equation: $y = (x + a)^2 + b,$ we get: $y = (x - 8)^2 - 3.$

Final Answer:

• $a = -8$,
• $b = -3$.

Would you like me to explain this further or solve related problems?

5 Related Questions:

1. How do you determine whether a quadratic curve opens upward or downward?
2. What is the general form of a quadratic equation, and how does it relate to this vertex form?
3. Can you find the equation of a parabola passing through other specific points?
4. How does completing the square help in converting a quadratic equation to vertex form?
5. What transformations are applied to $y = x^2$ to produce $y = (x - 8)^2 - 3$?

Tip:

The vertex form of a quadratic equation, $y = (x + a)^2 + b$, directly reveals the vertex as $(-a, b)$, making it useful for problems involving minimum or maximum points.