Tangent line to x²−2x+3 at x = 2

Given the function f(x) = x² − 2x + 3, find the slope of the tangent line to the curve at x = 2 and write the equation of the tangent line at that point.

Step-by-step

1. f(x) = 1x² − 2x + 3
   First identify the quadratic coefficient, linear coefficient, and constant term of the function, paying attention to signs.
2. f′(x) = 2x − 2
   Power rule for derivatives: the derivative of x² is 2x, of x is 1, and of a constant is 0
3. k = f′(2) = 2 × 2 − 2 = 2
   The value of the derivative at a point equals the slope of the tangent line there; this is the geometric meaning of the derivative.
4. y₀ = f(2) = 3
   Substitute the x-coordinate of the tangency point back into the original function to get its y-coordinate.
5. y − (3) = 2(x − 2) ⇒ y = 2x − 1
   Write the tangent line equation in point-slope form: slope comes from the derivative, point is the tangency point.

Why it works

A tangent line equation requires two things: the slope and the point of tangency, neither can be omitted. The slope is given by the value of the derivative function at the x-coordinate of the point of tangency. The y-coordinate of the point of tangency must be found by substituting the x-coordinate back into the original function, not into the derivative function. After obtaining these two pieces, write the equation using the point-slope form and then rearrange it into slope-intercept form. Substituting the point of tangency back into the original function is the step most easily missed in this problem.

Common mistakes

• Calculating the y-coordinate of the point of tangency using the derivative function instead of the original function.
• Writing a plus sign instead of a minus sign for the x-coordinate in the point-slope form.
• Only writing the slope and treating it as the tangent line equation, omitting the intercept.

Related

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