Find Increasing Intervals for y = x² + 10x + 16: Quadratic Function Analysis

To find the intervals where the function $y = x^2 + 10x + 16$ is increasing, we follow these steps:

• Step 1: Find the Vertex of the Parabola

The vertex of the parabola $y = ax^2 + bx + c$ is located at $x = -\frac{b}{2a}$. For our function, $a = 1$ and $b = 10$. Thus:

$x = -\frac{b}{2a} = -\frac{10}{2 \times 1} = -5$

• Step 2: Determine the Sign of the Derivative

To find where the function is increasing, compute the derivative of $y$:

$y' = \frac{d}{dx}(x^2 + 10x + 16) = 2x + 10$

The function is increasing where $y' > 0$:

$2x + 10 > 0$

Solving the inequality:

$2x > -10$

$x > -5$

• Step 3: Conclusion

The function $y = x^2 + 10x + 16$ is increasing for the interval $x > -5$.

Therefore, the solution to the problem is $x > -5$.