find the tangent lines that can touch f(x)=(x-1)^2 and g(x)=-x^2-2x at the same time and draw the graph

The graph above shows the functions \( f(x) = (x-1)^2 \) and \( g(x) = -x^2 - 2x \), along with the tangent line that touches both curves at the same time. The tangent line intersects \( f(x) \) at \( x_1 = \frac{1}{2} \) and \( g(x) \) at \( x_2 = -\frac{1}{2} \). The slope of this common tangent line is the same at both points, ensuring that the line is indeed tangent to both functions.

Would you like more details on any step, or do you have any questions?

Here are some related questions to further explore this topic:

1. How would the tangent line change if the functions were shifted vertically?
2. What would happen if the functions had different curvatures (e.g., one concave up, the other concave down)?
3. Can a cubic function and a quadratic function share a common tangent line? If so, how would you find it?
4. How does the discriminant of a quadratic equation relate to the number of tangent lines a quadratic function can have?
5. What geometric significance does the midpoint between the tangency points have?

**Tip:** When solving problems involving tangents to multiple curves, always ensure the slopes are equal, as this is a critical condition for tangency.

Learn how to find tangent lines that touch both f(x)=(x-1)^2 and g(x)=-x^2-2x simultaneously. This problem involves using calculus to determine points of tangency and ensuring slopes match for both functions. Suitable for advanced high school students.

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