To find the minimum value of the quadratic function $f(x) = x^2 + 7x + 15.3$, we can use the following approach:

1. Identify the coefficients:

The quadratic function is in the form $f(x) = ax^2 + bx + c$, where:

• $a = 1$ (coefficient of $x^2$)
• $b = 7$ (coefficient of $x$)
• $c = 15.3$ (constant term)

2. Find the vertex:

The vertex of a parabola given by $f(x) = ax^2 + bx + c$ occurs at $x = -\frac{b}{2a}$.

So, let's calculate the $x$-coordinate of the vertex: $x = -\frac{b}{2a} = -\frac{7}{2 \times 1} = -\frac{7}{2} = -3.5$

3. Calculate the minimum value:

Substitute $x = -3.5$ back into the function $f(x)$ to find the minimum value: $f(-3.5) = (-3.5)^2 + 7(-3.5) + 15.3$ $f(-3.5) = 12.25 - 24.5 + 15.3 = 3.05$

4. Conclusion:

The minimum value of the function $f(x) = x^2 + 7x + 15.3$ is 3.05 to the nearest hundredth.

Would you like more details or have any questions?

Here are some related questions:

1. How do you find the vertex of any quadratic function?
2. What does the vertex of a parabola represent in real-world contexts?
3. How can you determine if a quadratic function has a maximum or minimum value?
4. What are the properties of parabolas that open upwards versus downwards?
5. How does the discriminant of a quadratic equation relate to the graph of the function?

Tip: The vertex of a quadratic function gives the maximum or minimum value of the function, depending on the direction the parabola opens (upwards for a minimum, downwards for a maximum).