The gradient of the tangent to the curve at point A is -3.
Find
(i)
the coordinates of A
the equation of the tangent to the curve at point A

(il) Give your answer in the form y = mx + c.

To solve this problem, we need to follow these steps:

### Step 1: Find the coordinates of Point A
We know that the gradient of the tangent to the curve at point A is -3. This implies that the derivative of the curve's equation at point A, \( f'(x_A) \), is -3.

Assuming the curve is given by a function \( y = f(x) \), we'll need to know this function to differentiate it and set the derivative equal to -3. However, since the function is not provided, the specific coordinates of A cannot be determined directly. Typically, you'd need the equation of the curve to proceed.

### Step 2: Find the Equation of the Tangent to the Curve at Point A
The general equation of a tangent line to a curve at a point \( (x_A, y_A) \) is given by:
\[
y - y_A = m(x - x_A)
\]
where \( m \) is the gradient of the tangent, and \( (x_A, y_A) \) are the coordinates of point A.

Given that the gradient \( m \) is -3, the equation of the tangent line can be written as:
\[
y - y_A = -3(x - x_A)
\]

Rearranging this into the form \( y = mx + c \), we get:
\[
y = -3x + 3x_A + y_A
\]
So, the equation of the tangent is:
\[
y = -3x + c
\]
where \( c = 3x_A + y_A \).

### Conclusion
Without the actual function or additional information, we can’t determine the exact coordinates of point A or the specific value of \( c \) in the tangent's equation. But the general forms would be:

(i) **Coordinates of A**: Undetermined without the curve equation.

(ii) **Equation of the Tangent**: \( y = -3x + c \), where \( c \) is determined by the specific point A.

Would you like more details or have any questions?

### Relative Questions:
1. What additional information would allow us to find the coordinates of point A?
2. How does the derivative of a function relate to the gradient of the tangent?
3. Can the gradient of a tangent line be used to determine if a function is increasing or decreasing at a point?
4. What is the significance of the y-intercept in the equation of a line?
5. How would the equation change if the gradient were positive?

### Tip:
The slope of the tangent line to a curve at a point is always equal to the derivative of the function at that point.

Learn how to find the coordinates of point A and the equation of the tangent line to a curve with a given gradient of -3 at point A. Understand the relationship between derivatives and tangent gradients in calculus. Suitable for advanced high school students studying calculus.

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