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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.

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

Solve for a, b, and c in a quadratic equation given its minimum point at (4, -3) and passing through (5, 0). Learn to use the vertex form and convert it to standard form.

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