The problem asks to find the values of $a$, $b$, and $c$ in the quadratic equation:

$y = ax^2 + bx + c$

given the following conditions:

1. The graph has a minimum at the point $(5, -3)$.
2. The graph passes through the point $(4, 0)$.

Step 1: Use the vertex form of a parabola

The minimum point $(5, -3)$ tells us that the vertex of the parabola is at $x = 5$. The vertex form of a quadratic function is:

$y = a(x - h)^2 + k$

where $(h, k)$ is the vertex. In this case, $h = 5$ and $k = -3$, so we can write the equation as:

$y = a(x - 5)^2 - 3$

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

Now, substitute $x = 4$ and $y = 0$ (the point $(4, 0)$) into the equation:

$0 = a(4 - 5)^2 - 3$ $0 = a(1)^2 - 3$ $a - 3 = 0$ $a = 3$

Step 3: Write the equation in standard form

Now that we know $a = 3$, we can substitute this value back into the vertex form:

$y = 3(x - 5)^2 - 3$

Expand the equation:

$y = 3(x^2 - 10x + 25) - 3$ $y = 3x^2 - 30x + 75 - 3$ $y = 3x^2 - 30x + 72$

Thus, the quadratic equation is:

$y = 3x^2 - 30x + 72$

Conclusion

The values of $a$, $b$, and $c$ are:

• $a = 3$
• $b = -30$
• $c = 72$

Would you like any further details or explanations?

Related Questions:

1. How do we know the point $(5, -3)$ represents a minimum for this parabola?
2. What other forms of quadratic equations are useful for solving vertex problems?
3. How can you verify if a point lies on the graph of a quadratic function?
4. What is the significance of the discriminant in quadratic equations?
5. How can we find the axis of symmetry of any quadratic function?

Tip:

When given the vertex of a quadratic function, using the vertex form of the equation can make solving much simpler!