It looks like you're asking to evaluate or analyze the function $y = x^2 + 6x$ at the point $(-2, -8)$ using some parameters, though the parameters weren't mentioned in your message.

We can proceed with the following steps if the task is to confirm whether $(-2, -8)$ is on the curve, or find derivatives, tangent lines, or other parameters at that point. Here's a breakdown:

Step 1: Check if $(-2, -8)$ is on the curve.

We'll plug $x = -2$ into the equation and see if the result equals $y = -8$.

Given: $y = x^2 + 6x$ Substitute $x = -2$:

= 4 - 12 = -8$$ So, the point $$(-2, -8)$$ is on the curve. ### Step 2: Derivative of the function $$y = x^2 + 6x$$ The derivative represents the slope of the tangent line at any point on the curve. Using basic rules of differentiation: $$\frac{dy}{dx} = 2x + 6$$ Now substitute $$x = -2$$ into the derivative: $$\frac{dy}{dx} = 2(-2) + 6 = -4 + 6 = 2$$ Thus, the slope of the tangent line at $$(-2, -8)$$ is 2. ### Step 3: Equation of the tangent line at $$(-2, -8)$$ The slope of the tangent line is 2, and it passes through the point $$(-2, -8)$$. The equation of the tangent line is given by: $$y - y_1 = m(x - x_1)$$ Where $$m$$ is the slope and $$(x_1, y_1) = (-2, -8)$$. Substitute the values: $$y - (-8) = 2(x - (-2))$$ Simplifying: $$y + 8 = 2(x + 2)$$ $$y + 8 = 2x + 4$$ $$y = 2x - 4$$ Thus, the equation of the tangent line at $$(-2, -8)$$ is $$y = 2x - 4$$. ### Summary: 1. The point $$(-2, -8)$$ is on the curve. 2. The derivative (slope of the tangent line) at $$(-2, -8)$$ is 2. 3. The equation of the tangent line at $$(-2, -8)$$ is $$y = 2x - 4$$. Would you like more details on any of the steps or have other questions? Here are five related questions: 1. How do we find the second derivative of the function? 2. How do we calculate the normal line at $$(-2, -8)$$? 3. What is the concavity of the function near the point $$(-2, -8)$$? 4. How do we find the y-intercept of the tangent line? 5. What is the slope of the curve at different values of $$x$$? **Tip**: The first derivative gives the slope of the tangent line, while the second derivative reveals the concavity (how the curve bends) at a point.