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Let's work through each part of the problem based on the given function \( f(x) = -x^2 - 2x + 3 \).

### a. Find the derivative \( f'(x) \), using the definition of the derivative.
The derivative of \( f(x) \) can be found by using standard differentiation rules rather than the limit definition directly:
\[
f'(x) = \frac{d}{dx} \left( -x^2 - 2x + 3 \right)
= -2x - 2.
\]

### b. Find the slope of the tangent line to the graph of \( f \) at the point \( (0, 3) \).
The slope of the tangent line at a specific point is the value of the derivative at that point. We already found that \( f'(x) = -2x - 2 \). To find the slope at \( x = 0 \):
\[
f'(0) = -2(0) - 2 = -2.
\]
Thus, the slope of the tangent line at \( (0, 3) \) is \( -2 \).

### c. Find the rate of change of \( f \) when \( x = 0 \).
This is the same as finding the derivative at \( x = 0 \), which we already did in part (b). The rate of change at \( x = 0 \) is \( -2 \).

### d. Find an equation of the tangent line to the graph of \( f \) at the point \( (0, 3) \).
The equation of the tangent line is given by the point-slope form of a line:
\[
y - y_1 = m(x - x_1),
\]
where \( m \) is the slope and \( (x_1, y_1) \) is the point on the curve. We know the slope is \( -2 \) and the point is \( (0, 3) \), so:
\[
y - 3 = -2(x - 0),
\]
which simplifies to:
\[
y = -2x + 3.
\]
This is the equation of the tangent line.

### e. Sketch the graph of \( f \) and the tangent line to the curve at the point \( (0, 3) \).
The function \( f(x) = -x^2 - 2x + 3 \) is a downward-opening parabola (since the coefficient of \( x^2 \) is negative), and the tangent line has equation \( y = -2x + 3 \), which touches the parabola at the point \( (0, 3) \).

Would you like a visual sketch of this graph and the tangent line?

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Here are 5 follow-up questions for further practice:
1. What is the second derivative \( f''(x) \) of the function?
2. Find the vertex of the parabola \( f(x) = -x^2 - 2x + 3 \).
3. For which values of \( x \) is the function increasing or decreasing?
4. How does the concavity of the graph of \( f \) relate to the second derivative?
5. What are the x-intercepts of the graph of \( f(x) \)?

**Tip**: The slope of the tangent line to a curve at any point gives the instantaneous rate of change of the function at that point.

Let f(x) = -x^2 - 2x + 3. a) Find the derivative f'(x), b) Find the slope of the tangent line at (0, 3), c) Find the rate of change when x = 0, d) Find the equation of the tangent line at (0, 3), e) Sketch the graph of f and the tangent line at (0, 3).

Learn how to find the derivative, tangent line, and rate of change for the function f(x) = -x^2 - 2x + 3. This problem covers key calculus concepts, including derivatives and the point-slope formula. Suitable for students in Grades 10-12.

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