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)=x²+1; Q(3, 6)

Verified step by step guidance

Step 1: Find the derivative of the function f(x) = x^2 + 1 to determine the slope of the tangent line at any point P on the graph. The derivative, f'(x), represents the slope of the tangent line.

Step 2: Calculate f'(x) by differentiating f(x) = x^2 + 1. The derivative is f'(x) = 2x.

Step 3: Let P be a point (a, f(a)) on the graph of f. The slope of the tangent line at P is f'(a) = 2a. The equation of the tangent line at P is y - f(a) = 2a(x - a).

Step 4: Since the tangent line passes through Q(3, 6), substitute x = 3 and y = 6 into the tangent line equation: 6 - (a^2 + 1) = 2a(3 - a).

Step 5: Solve the equation 6 - (a^2 + 1) = 2a(3 - a) for a to find the x-coordinates of the points P. Substitute these values back into f(x) to find the corresponding y-coordinates.

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Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

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 equal to the derivative of the function at that point. For the function f(x) = x² + 1, the derivative f'(x) = 2x 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 the 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 from the derivative and the coordinates of point Q.

Graphing Functions and Tangents

Graphing the function and its tangent lines helps visualize the relationship between the function and the points of tangency. By plotting f(x) = x² + 1 and the tangent lines at various points P, we can confirm whether these lines intersect the point Q(3, 6). This graphical approach aids in verifying the algebraic solutions obtained.

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