The problem asks for the values of $a$, $b$, and $c$ in the quadratic equation $y = ax^2 + bx + c$, given that:

• The minimum point of the parabola occurs at $(5, -3)$.
• The graph passes through the point $(4, 0)$.

Step 1: Use the vertex form of a quadratic equation

Since the minimum (or vertex) is at $(5, -3)$, we can first express the equation of the parabola in vertex form: $y = a(x - h)^2 + k$ where $(h, k)$ is the vertex of the parabola. Substituting $h = 5$ and $k = -3$, we get: $y = a(x - 5)^2 - 3$

Step 2: Use the point $(4, 0)$

Now, we know that the parabola passes through $(4, 0)$. Substituting $x = 4$ and $y = 0$ into the equation: $0 = a(4 - 5)^2 - 3$ $0 = a(1)^2 - 3$ $0 = a - 3$ Thus, $a = 3$.

Step 3: Rewrite the equation in standard form

Now that we know $a = 3$, we can rewrite the equation in its expanded (standard) form: $y = 3(x - 5)^2 - 3$ Expanding this: $y = 3(x^2 - 10x + 25) - 3$ $y = 3x^2 - 30x + 75 - 3$ $y = 3x^2 - 30x + 72$

Step 4: Identify $a$, $b$, and $c$

From the expanded form, we have: $a = 3, \quad b = -30, \quad c = 72$

Thus, the values of $a$, $b$, and $c$ are:

• $a = 3$
• $b = -30$
• $c = 72$

Would you like more details on any of these steps? Here are some related questions you might find useful:

1. How do you find the vertex form of a quadratic equation?
2. How can the axis of symmetry help in finding the vertex of a parabola?
3. How does completing the square relate to converting from standard to vertex form?
4. What are the key features of a parabola derived from the quadratic equation?
5. Can you use the quadratic formula to verify the vertex?

Tip: Remember that the vertex of a parabola given by $ax^2 + bx + c$ occurs at $x = -\frac{b}{2a}$. This can be a quick way to identify the x-coordinate of the minimum or maximum point.