We are given the quadratic function $y = ax^2 + bx + c$ with the following information:

1. The minimum point of the parabola is at $(4, -3)$.
2. The parabola passes through the point $(5, 0)$.

Step 1: Use the vertex form of a quadratic equation.

The vertex form of a quadratic equation is given by: $y = a(x - h)^2 + k$ where $(h, k)$ is the vertex. From the problem, we know that the vertex (minimum point) is $(4, -3)$, so: $y = a(x - 4)^2 - 3$

Step 2: Use the point $(5, 0)$ to find $a$.

Substitute the point $(5, 0)$ into the equation to find $a$: $0 = a(5 - 4)^2 - 3$ $0 = a(1)^2 - 3$ $0 = a - 3$ $a = 3$

Step 3: Substitute $a = 3$ into the vertex form equation.

Now that we know $a = 3$, the equation becomes: $y = 3(x - 4)^2 - 3$

Step 4: Expand the equation to the standard form.

To express the equation in the form $y = ax^2 + bx + c$, expand the vertex form: $y = 3(x - 4)^2 - 3$ First expand $(x - 4)^2$: $y = 3(x^2 - 8x + 16) - 3$ Distribute the 3: $y = 3x^2 - 24x + 48 - 3$ Simplify: $y = 3x^2 - 24x + 45$

Step 5: Identify $a$, $b$, and $c$.

From the equation $y = 3x^2 - 24x + 45$, we can see that:

• $a = 3$
• $b = -24$
• $c = 45$

Thus, the values of $a$, $b$, and $c$ are: $a = 3, \quad b = -24, \quad c = 45$

Would you like a further breakdown of the steps or any additional details? Here are some related questions:

1. How is the vertex form of a quadratic equation derived?
2. Why is the point $(4, -3)$ considered the vertex?
3. Can you explain how to convert from vertex form to standard form?
4. What does the value of $a$ tell us about the shape of the parabola?
5. Could the minimum point change if we had different values for $a$, $b$, and $c$?

Tip: Always substitute known points into an equation to find unknown variables before expanding or simplifying.