To determine the intervals of increase and decrease for the function $f(x) = x^2 + 8x - 7$, let's follow these steps:

Step 1: Differentiate the function

Find the derivative of $f(x)$: $f'(x) = 2x + 8$

Step 2: Solve for critical points

Set $f'(x) = 0$ to find the critical points: $2x + 8 = 0 \implies x = -4$

Step 3: Test intervals around the critical point

Divide the domain into intervals based on the critical point $x = -4$. The intervals are:

1. $(-\infty, -4)$
2. $(-4, \infty)$

Test the sign of $f'(x)$ in each interval:

• For $x \in (-\infty, -4)$: Choose $x = -5$, $f'(-5) = 2(-5) + 8 = -10 + 8 = -2 \quad (\text{negative, so } f(x) \text{ is decreasing})$
• For $x \in (-4, \infty)$: Choose $x = 0$, $f'(0) = 2(0) + 8 = 8 \quad (\text{positive, so } f(x) \text{ is increasing})$

Step 4: Determine intervals of increase and decrease

• The function $f(x)$ is decreasing on $(-\infty, -4)$.
• The function $f(x)$ is increasing on $(-4, \infty)$.

Would you like a detailed graph or additional explanation?

Related Questions:

1. What are the coordinates of the minimum or maximum point of this function?
2. Can we determine whether this function has a vertex using completing the square?
3. How do we verify intervals of monotonicity using second derivatives?
4. How can we sketch this parabola for better understanding?
5. What is the domain and range of this function?

Tip: Always test the derivative in specific intervals to confirm monotonicity changes.