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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.

The graph of y = ax^2 + bx + c has a minimum at (5, -3) and passes through (4, 0). Find the values of a, b, and c.

Learn how to find the coefficients a, b, and c for a quadratic equation y = ax^2 + bx + c given a minimum at (5, -3) and the graph passing through (4, 0). This solution involves using the vertex form and standard form of a quadratic equation, suitable for students in Grades 9-11.

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