Worksheets · Precalculus

Precalculus tangent lines worksheet

In Precalculus the slope of a tangent line comes from the limit of secant slopes, not a derivative rule. Find that limit, then use the point on the curve to write the line in point-slope form.

Also taught in Calculus, at its own level and with its own examples.

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Pick more than one for a sheet that mixes them.

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Free math worksheets at sigmaprep.io/worksheets© 2026 Sigma Prep

Sigma Prep · sigmaprep.io/worksheets

Name

Tangent and Normal Lines

Date Period

Find each line or point.

  1. Tangent to y=4x3+4x2+6x at x=0
  2. Normal to y=−x2−4 at x=−2
  3. Tangent to y=−3sin⁡x at x=π
  4. Find the x-values on y=x3−3x where the tangent is parallel to y=9x−4
  5. Tangent to y=−x3+6x2 at x=−1
  6. Normal to y=4x2 at x=3
  7. Tangent to y=−ex at x=0
  8. Find the point on y=2x2+6x where the tangent is parallel to y=2x+2
  9. Tangent to y=x3−7x2+5x−4 at x=3
  10. Normal to y=−4x2−x−6 at x=−3
  11. Tangent to y=−4x at x=9
  12. Find the point on y=x2−4x where the tangent is parallel to y=−12x−2
  13. Tangent to y=−2x3 at x=−3
  14. Normal to y=−3x2+3 at x=3
  15. Tangent to y=2x at x=4
  16. Find the point on y=x2−4x where the tangent is parallel to y=8x−5
  17. Tangent to y=3x2+5x−5 at x=1
  18. Normal to y=−4x3+8x2+3 at x=−4
  19. Tangent to y=3sin⁡x at x=π
  20. Find the point on y=2x2−5x where the tangent is parallel to y=−9x−1

Free math worksheets at sigmaprep.io/worksheets© 2026 Sigma Prep

Sigma Prep · sigmaprep.io/worksheets

Name

Tangent and Normal LinesAnswer keyVersion 1

Date Period

Find each line or point.

  1. Tangent to y=4x3+4x2+6x at x=0y=6x
  2. Normal to y=−x2−4 at x=−2y+8=−14(x+2)
  3. Tangent to y=−3sin⁡x at x=πy=3x−3π
  4. Find the x-values on y=x3−3x where the tangent is parallel to y=9x−4x=−2,x=2
  5. Tangent to y=−x3+6x2 at x=−1y=−15x−8
  6. Normal to y=4x2 at x=3y−36=−124(x−3)
  7. Tangent to y=−ex at x=0y=−x−1
  8. Find the point on y=2x2+6x where the tangent is parallel to y=2x+2(−1,−4)
  9. Tangent to y=x3−7x2+5x−4 at x=3y=−10x+5
  10. Normal to y=−4x2−x−6 at x=−3y+39=−123(x+3)
  11. Tangent to y=−4x at x=9y=−23x−6
  12. Find the point on y=x2−4x where the tangent is parallel to y=−12x−2(−4,32)
  13. Tangent to y=−2x3 at x=−3y=−54x−108
  14. Normal to y=−3x2+3 at x=3y+24=118(x−3)
  15. Tangent to y=2x at x=4y=12x+2
  16. Find the point on y=x2−4x where the tangent is parallel to y=8x−5(6,12)
  17. Tangent to y=3x2+5x−5 at x=1y=11x−8
  18. Normal to y=−4x3+8x2+3 at x=−4y−387=1256(x+4)
  19. Tangent to y=3sin⁡x at x=πy=−3x+3π
  20. Find the point on y=2x2−5x where the tangent is parallel to y=−9x−1(−1,7)

Free math worksheets at sigmaprep.io/worksheets© 2026 Sigma Prep

How to do these

Tangent to f(x)=1x at x=2

  1. Secant slope: (1/(2 + h) - 1/2)/h simplifies to -1/(2(2 + h)).
  2. As h goes to 0 the slope is -1/4.
  3. The point is (2, 1/2): y - 1/2 = -1/4(x - 2).

The answer is y=−14x+1.

Where students go wrong

Using a secant slope as the answer. A secant through two nearby points only estimates the slope; the tangent needs the limit as h goes to 0.

Also called equation of a tangent line, normal line or horizontal tangent.

What you can put on this worksheet

Tangent lines
Tangent to y=x2+2 at x=−1 → y=−2x+1
Normal lines
Normal to y=x2+2 at x=−1 → y−3=12(x+1)
Where the tangent is horizontal
Where is the tangent horizontal? y=x3+6x2+3 → x=−4, x=0
Trig, exponential, log, roots and reciprocals
Tangent to y=5e−2x at x=0 → y=−10x+5
Parallel to a given line
Find the point on y=2x2 where the tangent is parallel to y=4x+2 → (1,2)

Questions about these worksheets

Yes. Every tangent lines sheet is free to print and free to download as a PDF. There is no account to make, no email to hand over and no limit on how many you take.

Yes. The answer key prints on a second page, and you can choose whether it shows each question beside its answer or just the answers in a list.

Yes. Print straight from the page, or download the PDF and print that. The sheet is laid out for paper rather than squeezed off a screen, so there is room to work under each question, and the answer key comes out on its own page.

Precalculus is usually taken around 11th or 12th grade, though schools vary and plenty of students meet it earlier or later. Pick the difficulty rather than the grade: the easy level suits a first lesson on it, the hard level suits review before a test.

Yes, as many as you like. The questions are built fresh each time rather than picked from a fixed set of files, so pressing Generate gives a new sheet. That is what you want for a second class, a retake, or two students sitting next to each other.

That is the closest comparison, and Kuta Software is good software. It is also paid software you install on a computer. These worksheets run in a browser tab for free, with no account and nothing to install. Like Kuta, the tangent lines questions are generated when you ask for them rather than pulled from a fixed set of files, so two classes never get the same sheet, and the answer key comes with it.

Yes. There are three levels, and you can tick more than one for a sheet that mixes them, in which case the questions come out easiest first. The harder levels are not just larger numbers: they ask for something the easy ones do not.

You pick from 5: tangent lines, normal lines, where the tangent is horizontal, trig, exponential, log, roots and reciprocals and parallel to a given line. Tick as many as you want and set how many of each, or let it spread them evenly.

Using the derivative as the y value. It gives the slope; the point still comes from the original function, and mixing the two puts the line in the wrong place.