at what point on the graph of f(x) = x^2-4 is the tangent line parallel to the line 6x-3y=2

To find the point on the graph of \( f(x) = x^2 - 4 \) where the tangent line is parallel to the line \( 6x - 3y = 2 \), follow these steps:

### Step 1: Determine the slope of the given line.
The equation of the line is \( 6x - 3y = 2 \). To find the slope, rewrite this equation in slope-intercept form \( y = mx + b \), where \( m \) is the slope.

\[
6x - 3y = 2
\]

Divide the entire equation by 3:

\[
2x - y = \frac{2}{3}
\]

Now, solve for \( y \):

\[
y = 2x - \frac{2}{3}
\]

So, the slope \( m \) of the line is \( 2 \).

### Step 2: Find the derivative of \( f(x) \).
The derivative of \( f(x) = x^2 - 4 \) gives the slope of the tangent line at any point \( x \).

\[
f'(x) = 2x
\]

### Step 3: Set the derivative equal to the slope of the line.
We want the slope of the tangent line \( f'(x) \) to be equal to 2 (the slope of the given line).

\[
2x = 2
\]

Solve for \( x \):

\[
x = 1
\]

### Step 4: Find the corresponding \( y \)-coordinate.
Substitute \( x = 1 \) back into the original function \( f(x) = x^2 - 4 \) to find the \( y \)-coordinate.

\[
y = 1^2 - 4 = 1 - 4 = -3
\]

### Conclusion:
The point on the graph of \( f(x) = x^2 - 4 \) where the tangent line is parallel to the line \( 6x - 3y = 2 \) is \( (1, -3) \).

Would you like further details or have any questions?

Here are 5 related questions:
1. How do you find the slope of a tangent line to any curve?
2. What does it mean for two lines to be parallel in terms of their slopes?
3. How do you find the equation of the tangent line at a given point on a curve?
4. What is the significance of the derivative in determining the slope of a curve?
5. Can a function have more than one tangent line parallel to a given line?

**Tip:** Remember that the slope of a tangent line at any point on a curve is given by the derivative of the function at that point.

Learn how to find the point on the graph of f(x) = x^2 - 4 where the tangent line is parallel to the line 6x - 3y = 2. Understand the derivative of f(x), slope determination, and solve using algebraic methods. Suitable for Grades 10-12.

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