let k be the y intercept of the tangent line to the graph of f(x)=x^2 +x passing through point (1,-2). find k

Let's find the y-intercept, \( k \), of the tangent line to the graph of \( f(x) = x^2 + x \) passing through the point \( (1, -2) \).

1. **Find the derivative of \( f(x) \)**:
   \[
   f'(x) = 2x + 1
   \]
   The derivative represents the slope of the tangent line at any point \( x \).

2. **Find the slope of the tangent line at \( x = 1 \)**:
   \[
   f'(1) = 2(1) + 1 = 3
   \]
   So, the slope of the tangent line at \( x = 1 \) is 3.

3. **Equation of the tangent line**:
   The equation of a line with slope \( m \) passing through a point \((x_0, y_0)\) is given by:
   \[
   y - y_0 = m(x - x_0)
   \]
   Using the point \( (1, -2) \) and slope \( 3 \):
   \[
   y - (-2) = 3(x - 1)
   \]
   Simplifying:
   \[
   y + 2 = 3x - 3
   \]
   \[
   y = 3x - 5
   \]

4. **Find the y-intercept \( k \)**:
   The y-intercept is the value of \( y \) when \( x = 0 \):
   \[
   k = 3(0) - 5 = -5
   \]

Thus, the y-intercept \( k \) of the tangent line is \( \boxed{-5} \).

Would you like to explore more details or have any questions? Here are some related questions:

1. How do you find the equation of a tangent line for a general function at a given point?
2. What is the geometric interpretation of a derivative at a point?
3. How do you find the y-intercept of a line given its equation?
4. What are the conditions for a line to be tangent to a curve?
5. How do we verify if a given line is tangent to a specific curve?

**Tip**: For any curve \( f(x) \), the tangent line at a point \( (a, f(a)) \) can be found using the derivative \( f'(a) \) as the slope of the line.

Let k be the y-intercept of the tangent line to the graph of f(x) = x^2 + x passing through point (1, -2). Find k.

Learn how to find the y-intercept of the tangent line to the function f(x) = x^2 + x passing through the point (1, -2). This problem covers key calculus concepts such as derivatives and tangent lines. Suitable for students in Grades 10-12.

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