Given the function f and the point Q, find all points P on the graph of f such that the line tangent to f at P passes through Q. Check your work by graphing f and the tangent lines.
f(x) = 1/x; Q (-2, 4)

Verified step by step guidance

1. Step 1: Identify the point P on the graph of f(x) = \frac{1}{x} as (a, f(a)) = (a, \frac{1}{a}).
2. Step 2: Find the derivative of f(x) = \frac{1}{x}, which is f'(x) = -\frac{1}{x^2}. This represents the slope of the tangent line at any point P.
3. Step 3: Write the equation of the tangent line at point P using the point-slope form: y - \frac{1}{a} = -\frac{1}{a^2}(x - a).
4. Step 4: Substitute the coordinates of point Q (-2, 4) into the tangent line equation: 4 - \frac{1}{a} = -\frac{1}{a^2}(-2 - a).
5. Step 5: Solve the equation from Step 4 for a to find the x-coordinates of all points P where the tangent line passes through Q.

Key Concepts

Tangent Line

A tangent line to a function at a given point is a straight line that touches the graph of the function at that point without crossing it. The slope of the tangent line is determined by the derivative of the function at that point. For the function f(x) = 1/x, the derivative f'(x) = -1/x^2 gives the slope of the tangent line at any point P on the graph.

Finding Points on a Graph

To find points P on the graph of a function where a tangent line passes through a specific point Q, we need to set up an equation that relates the coordinates of P and Q. This involves using the point-slope form of the line equation, which incorporates the slope of the tangent line and the coordinates of point P. Solving this equation will yield the x-coordinates of the points P.

Graphing Functions and Tangents

Graphing the function f(x) = 1/x and its tangent lines helps visualize the relationship between the function and the points where the tangent lines intersect the point Q. By plotting the function and the calculated tangent lines, one can verify the accuracy of the solutions found algebraically. This graphical representation aids in understanding the behavior of the function and the tangents at various points.