Equation of a tangent line

A tangent line is the line with slope $f'(a)$ through the point $(a,\,f(a))$.

Get the slope by differentiating, plug in the coordinates of the point, and the equation is determined. Do not forget to compute the $y$-coordinate of the point of tangency from $f(a)$.

Examples

1. 01Tangent line at $x=2$. What goes in □?$\displaystyle f(x)=x^{2} \;\Rightarrow\; y=\square(x-2)+4$AnswerHide

   Answer$\displaystyle 4$
2. 02What are the coordinates of the point of tangency at $x=1$?$\displaystyle f(x)=x^{2}$AnswerHide

   Answer$\displaystyle (1,\ 1)$
3. 03Tangent line at $x=-1$. What goes in □?$\displaystyle f(x)=x^{2} \;\Rightarrow\; y=\square(x+1)+1$AnswerHide

   Answer$\displaystyle -2$
4. 04Tangent line at $x=1$. What goes in □?$\displaystyle f(x)=x^{2} \;\Rightarrow\; y=\square(x-1)+1$AnswerHide

   Answer$\displaystyle 2$
5. 05Tangent line at $x=3$. What goes in □?$\displaystyle f(x)=x^{2} \;\Rightarrow\; y=\square(x-3)+9$AnswerHide

   Answer$\displaystyle 6$
6. 06Tangent line at $x=-2$. What goes in □?$\displaystyle f(x)=x^{2} \;\Rightarrow\; y=\square(x+2)+4$AnswerHide

   Answer$\displaystyle -4$

FAQ

Q1Where does the slope of the tangent come from?Show answerHide

A

From the derivative at the point of tangency, $f'(a)$. The steps are: find $f'(x)$, substitute $x=a$, then write the line from the point and the slope.

Practice this pattern in the app, 5 problems a day.

Published 2026-09-03 · Updated 2026-09-05