Find all points of the graph of f(x) = 3x^2 + 9x whose tangent lines are parallel to the line  y- 33x = 0.

To find all points on the graph of the function \( f(x) = 3x^2 + 9x \) where the tangent lines are parallel to the line \( y - 33x = 0 \), we need to follow these steps:

### Step 1: Determine the slope of the given line

The equation of the line \( y - 33x = 0 \) can be rewritten as:
\[
y = 33x
\]
Here, the slope of the line is \( 33 \), which means the tangent lines to the graph of \( f(x) \) that we are looking for must also have a slope of \( 33 \).

### Step 2: Find the derivative of \( f(x) \)

The derivative of a function gives the slope of the tangent line at any point on the graph. We compute the derivative of \( f(x) = 3x^2 + 9x \):
\[
f'(x) = \frac{d}{dx}(3x^2 + 9x) = 6x + 9
\]
This derivative represents the slope of the tangent line at any point \( x \) on the graph.

### Step 3: Set the derivative equal to 33

To find the points where the slope of the tangent line is 33, we set the derivative equal to 33:
\[
6x + 9 = 33
\]
Now, solve for \( x \):
\[
6x = 33 - 9
\]
\[
6x = 24
\]
\[
x = 4
\]

### Step 4: Find the corresponding \( y \)-coordinate

Now that we have \( x = 4 \), substitute this back into the original function \( f(x) \) to find the corresponding \( y \)-coordinate:
\[
f(4) = 3(4)^2 + 9(4) = 3(16) + 36 = 48 + 36 = 84
\]
Thus, the point is \( (4, 84) \).

### Final Answer:

The point on the graph of \( f(x) = 3x^2 + 9x \) where the tangent line is parallel to the line \( y - 33x = 0 \) is \( (4, 84) \).

Would you like more details or have any questions?

Here are 5 related questions to deepen your understanding:
1. What is the significance of the slope in determining parallel lines?
2. How would the solution change if the line equation was \( y - mx = 0 \)?
3. How do you interpret the derivative geometrically in this context?
4. How would you find tangent lines perpendicular to a given line instead?
5. How does the general form of a quadratic function affect its tangent lines?

**Tip:** The derivative of a function always represents the instantaneous rate of change or the slope of the tangent line at any specific point.

Find all points of the graph of f(x) = 3x^2 + 9x whose tangent lines are parallel to the line y - 33x = 0.

Explore how to find points on the quadratic function f(x) = 3x^2 + 9x where the tangent lines are parallel to the line y = 33x. Learn key concepts like differentiation and parallel line slopes. Suitable for Grades 10-12 students.

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