To determine all coordinate points $(x, y)$ on the graph of the function $f(x) = x^3 + 6x^2 + 7x - 2$ where the tangent line has a slope of $-2$, we need to follow these steps:

Step 1: Find the derivative of $f(x)$

The derivative of a function gives the slope of the tangent line at any point $x$. So, let's differentiate $f(x)$:

$f'(x) = \frac{d}{dx}(x^3 + 6x^2 + 7x - 2)$

Using the power rule, we get:

$f'(x) = 3x^2 + 12x + 7$

Step 2: Set the derivative equal to the slope $-2$

Since we are looking for the points where the slope of the tangent line is $-2$, set the derivative equal to $-2$:

$3x^2 + 12x + 7 = -2$

Step 3: Solve for $x$

Rearrange the equation to set it equal to zero:

$3x^2 + 12x + 7 + 2 = 0$

$3x^2 + 12x + 9 = 0$

Now, divide the whole equation by 3:

$x^2 + 4x + 3 = 0$

This is a quadratic equation. We can factor it as:

$(x + 3)(x + 1) = 0$

Thus, the solutions for $x$ are:

$x = -3 \quad \text{or} \quad x = -1$

Step 4: Find the corresponding $y$-coordinates

To find the corresponding $y$-coordinates, substitute $x = -3$ and $x = -1$ into the original function $f(x)$:

• For $x = -3$:

$f(-3) = (-3)^3 + 6(-3)^2 + 7(-3) - 2 = -27 + 54 - 21 - 2 = 4$

So, the point is $(-3, 4)$.

• For $x = -1$:

$f(-1) = (-1)^3 + 6(-1)^2 + 7(-1) - 2 = -1 + 6 - 7 - 2 = -4$

So, the point is $(-1, -4)$.

Step 5: Final answer

The two coordinate points where the slope of the tangent line is $-2$ are:

$(-3, 4) \quad \text{and} \quad (-1, -4)$

Would you like more details or have any questions?

Here are 5 related questions for further exploration:

1. How do you find the tangent line equation at a specific point on a curve?
2. What is the geometric meaning of the derivative of a function at a point?
3. How does the concavity of a function affect the shape of its graph?
4. How can you use derivatives to find local maxima and minima of a function?
5. What are the steps to solving cubic equations for their roots?

Tip: Remember, the derivative gives the instantaneous rate of change (or slope) of the function at any given point!