To find the range of the quadratic function $f(x) = 4x^2 + 4x + 3$, we can first rewrite it in vertex form by completing the square.

1. Identify the coefficients:

   • $a = 4$
   • $b = 4$
   • $c = 3$

2. Complete the square: $f(x) = 4(x^2 + x) + 3$ Take half of the coefficient of $x$ (which is $1$), square it ($\left(\frac{1}{2}\right)^2 = \frac{1}{4}$), and add and subtract this inside the parentheses: $f(x) = 4\left(x^2 + x + \frac{1}{4} - \frac{1}{4}\right) + 3$ Simplifying gives: $f(x) = 4\left(\left(x + \frac{1}{2}\right)^2 - \frac{1}{4}\right) + 3$ $f(x) = 4\left(x + \frac{1}{2}\right)^2 - 1 + 3$ $f(x) = 4\left(x + \frac{1}{2}\right)^2 + 2$

3. Determine the vertex: The vertex form $f(x) = 4\left(x + \frac{1}{2}\right)^2 + 2$ shows that the vertex is at $\left(-\frac{1}{2}, 2\right)$.

4. Analyze the parabola: Since $a = 4 > 0$, the parabola opens upwards. The minimum value of $f(x)$ occurs at the vertex.

5. Find the range: The minimum value is $2$, and since the parabola opens upwards, the range is: $[2, \infty)$

Conclusion

The range of the function $f(x) = 4x^2 + 4x + 3$ is $[2, \infty)$.

Do you want any details or have any questions? Here are some related questions you might consider:

1. How would you find the vertex of a different quadratic function?
2. What is the significance of the discriminant in a quadratic equation?
3. How do you determine if a quadratic function opens upward or downward?
4. Can you explain how to graph a quadratic function?
5. What are some applications of quadratic functions in real life?

Tip: When completing the square, always remember to balance the equation by adding and subtracting the same value to maintain equality.