Worksheet: Tangents to the Graph of a Function

Download Practice

In this worksheet, we will practice finding the slope and equation of the tangent to a curve at a given point using derivatives.

Q1:

Find the equation of the tangent to the curve 𝑦 = βˆ’ 2 π‘₯ + 8 π‘₯ βˆ’ 1 9   at π‘₯ = 2 .

  • A 𝑦 βˆ’ 4 π‘₯ βˆ’ 5 = 0
  • B βˆ’ 8 𝑦 + π‘₯ βˆ’ 2 = 0
  • C 𝑦 + 8 π‘₯ + 1 9 = 0
  • D 𝑦 βˆ’ 8 π‘₯ + 1 9 = 0

Q2:

If the line 𝑦 = 3 π‘₯ + 9 is tangent to the graph of the function 𝑓 at ( 2 , 1 5 ) , what is 𝑓 β€² ( 2 ) ?

Q3:

What is the π‘₯ -coordinate of the point where the tangent line to 𝑦 = π‘₯ + 1 2 π‘₯ + 1 1  is parallel to the π‘₯ -axis?

  • A βˆ’ 1 2
  • B6
  • C0
  • D βˆ’ 6

Q4:

The point ( 3 , 3 ) lies on the curve 𝑦 = 7 π‘₯ + π‘Ž π‘₯ + 𝑏  . If the slope of the tangent there is βˆ’ 1 , what are the values of constants π‘Ž and 𝑏 ?

  • A π‘Ž = βˆ’ 4 3 , 𝑏 = βˆ’ 1 7
  • B π‘Ž = βˆ’ 4 1 , 𝑏 = 6 3
  • C π‘Ž = 4 1 , 𝑏 = βˆ’ 1 8 3
  • D π‘Ž = βˆ’ 4 3 , 𝑏 = 6 9

Q5:

Find the point on the curve 𝑦 = βˆ’ 4 0 π‘₯ + 4 0  at which the tangent to the curve is parallel to the π‘₯ -axis.

  • A ( βˆ’ 1 , 0 )
  • B ( 0 , 0 )
  • C ( 1 , 0 )
  • D ( 0 , 4 0 )

Q6:

List the equations of all the tangents to 𝑦 = βˆ’ π‘₯  that also lie on the point ( 2 , βˆ’ 3 ) .

  • A 𝑦 βˆ’ 6 π‘₯ βˆ’ 9 = 0 , 𝑦 βˆ’ 2 π‘₯ βˆ’ 1 = 0
  • B 𝑦 βˆ’ 6 π‘₯ + 2 7 = 0 , 𝑦 βˆ’ 2 π‘₯ + 3 = 0
  • C 𝑦 + 6 π‘₯ + 2 7 = 0 , 𝑦 + 2 π‘₯ + 3 = 0
  • D 𝑦 + 6 π‘₯ βˆ’ 9 = 0 , 𝑦 + 2 π‘₯ βˆ’ 1 = 0

Q7:

Find the equation of the tangent to the curve 𝑦 = π‘₯ + 9 π‘₯ + 2 6 π‘₯   that makes an angle of 1 3 5 ∘ with the positive π‘₯ -axis.

  • A 𝑦 + 2 7 π‘₯ + 1 0 5 = 0
  • B 𝑦 βˆ’ 8 π‘₯ = 0
  • C 𝑦 βˆ’ π‘₯ 3 + 2 3 = 0
  • D 𝑦 + π‘₯ + 2 7 = 0

Q8:

Suppose the line 𝑦 + 5 π‘₯ βˆ’ 1 = 0 touches the curve 𝑓 ( π‘₯ ) = π‘₯ βˆ’ π‘₯ + π‘Ž  . What is π‘Ž ?

Q9:

If the curve 𝑦 = π‘Ž π‘₯ + 𝑏 π‘₯ + 2 π‘₯ + 7   is tangent to the line 𝑦 = 7 π‘₯ βˆ’ 3 at ( βˆ’ 1 , βˆ’ 1 0 ) , find the constants π‘Ž and 𝑏 .

  • A π‘Ž = βˆ’ 2 5 , 𝑏 = 4 0
  • B π‘Ž = βˆ’ 4 0 , 𝑏 = βˆ’ 2 5
  • C π‘Ž = 5 , 𝑏 = 1 0
  • D π‘Ž = βˆ’ 2 5 , 𝑏 = βˆ’ 4 0

Q10:

The line π‘₯ βˆ’ 𝑦 βˆ’ 3 = 0 touches the curve 𝑦 = π‘Ž π‘₯ + 𝑏 π‘₯   at ( 1 , βˆ’ 2 ) . Find π‘Ž and 𝑏 .

  • A π‘Ž = βˆ’ 1 3 , 𝑏 = βˆ’ 7
  • B π‘Ž = βˆ’ 7 , 𝑏 = 5
  • C π‘Ž = 1 3 , 𝑏 = βˆ’ 1 5
  • D π‘Ž = 5 , 𝑏 = βˆ’ 7

Q11:

Do the curves 𝑦 = βˆ’ 2 π‘₯ + 4 π‘₯ + 2 4  and 𝑦 = βˆ’ 6 π‘₯ βˆ’ 4 π‘₯ + 2 0  have a common tangent at the point of intersection? If so, give the equation of this tangent.

  • AYes, 𝑦 + 4 π‘₯ βˆ’ 1 4 = 0
  • BYes, 𝑦 βˆ’ 4 π‘₯ βˆ’ 2 2 = 0
  • CNo
  • DYes, 𝑦 βˆ’ 8 π‘₯ βˆ’ 2 6 = 0

Q12:

Determine the equation of the line tangent to the curve 𝑦 = 4 π‘₯ βˆ’ 2 π‘₯ + 4   at point ( βˆ’ 1 , βˆ’ 2 ) .

  • A 𝑦 = 1 4 π‘₯ + 1 2
  • B 𝑦 = 8 π‘₯ + 6
  • C 𝑦 = 1 6 π‘₯ βˆ’ 2
  • D 𝑦 = 1 6 π‘₯ + 1 4
  • E 𝑦 = 1 6 π‘₯ + 1 6

Q13:

The line 𝑦 + 2 π‘₯ + π‘Ž = 0 is tangent to the curve 𝑦 = π‘₯ βˆ’ 1  at the point ( 𝑏 , 𝑐 ) . Find π‘Ž , 𝑏 , and 𝑐 .

  • A π‘Ž = 4 , 𝑏 = βˆ’ 2 , 𝑐 = 3
  • B π‘Ž = βˆ’ 2 , 𝑏 = 1 , 𝑐 = 0
  • C π‘Ž = βˆ’ 4 , 𝑏 = 2 , 𝑐 = 3
  • D π‘Ž = 2 , 𝑏 = βˆ’ 1 , 𝑐 = 0

Q14:

The line 𝑦 = 5 π‘₯ + 4 is tangent to the graph of function 𝑓 at the point ( βˆ’ 1 , βˆ’ 1 ) . What is 𝑓 β€² ( βˆ’ 1 ) ?

Q15:

The line 5 π‘₯ + 𝑦 = 2 2 touches the curve 𝑦 = π‘Ž π‘₯ + 𝑏 π‘₯ βˆ’ 4 π‘₯ + 2 3   at the point ( 1 , 1 7 ) . Find π‘Ž and 𝑏 .

  • A π‘Ž = βˆ’ 7 , 𝑏 = 5
  • B π‘Ž = βˆ’ 5 , 𝑏 = 3
  • C π‘Ž = βˆ’ 7 , 𝑏 = βˆ’ 5
  • D π‘Ž = 3 , 𝑏 = βˆ’ 5

Q16:

Find the equation of the tangent to the curve 𝑦 = π‘₯ βˆ’ 2 π‘₯  at the point ( π‘₯ , 3 ) on the curve.

  • A 4 𝑦 βˆ’ π‘₯ βˆ’ 9 = 0 , βˆ’ 4 𝑦 βˆ’ π‘₯ + 1 1 = 0
  • B 𝑦 + 4 π‘₯ βˆ’ 1 5 = 0 , 𝑦 βˆ’ 4 π‘₯ βˆ’ 7 = 0
  • C βˆ’ 4 𝑦 βˆ’ π‘₯ + 1 5 = 0 , 4 𝑦 βˆ’ π‘₯ βˆ’ 1 3 = 0
  • D 𝑦 βˆ’ 4 π‘₯ + 9 = 0 , 𝑦 + 4 π‘₯ + 1 = 0

Q17:

Find the equation of the tangent to the curve 𝑦 = 8 π‘₯ + 5 π‘₯ βˆ’ 6  at the point ( βˆ’ 1 , βˆ’ 3 ) .

  • A βˆ’ 1 1 𝑦 βˆ’ π‘₯ βˆ’ 3 4 = 0
  • B 𝑦 βˆ’ 1 1 π‘₯ βˆ’ 8 = 0
  • C 1 1 𝑦 βˆ’ π‘₯ + 3 2 = 0
  • D 𝑦 + 1 1 π‘₯ + 1 4 = 0

Q18:

List the equations of the normals to 𝑦 = π‘₯ + 2 π‘₯  at the points where this curve meets the line 𝑦 βˆ’ 4 π‘₯ = 0 .

  • A 𝑦 βˆ’ 2 π‘₯ = 0 , 𝑦 βˆ’ 6 π‘₯ + 4 = 0
  • B 𝑦 + 2 π‘₯ = 0 , 𝑦 + 6 π‘₯ βˆ’ 4 = 0
  • C 2 𝑦 βˆ’ π‘₯ = 0 , 6 𝑦 βˆ’ π‘₯ βˆ’ 4 6 = 0
  • D 2 𝑦 + π‘₯ = 0 , 6 𝑦 + π‘₯ βˆ’ 5 0 = 0

Q19:

Determine the equation of the line normal to the curve 4 𝑦 = π‘₯  οŠͺ at the point ( 2 , βˆ’ 2 ) .

  • A 𝑦 = βˆ’ π‘₯ 2 βˆ’ 1
  • B 𝑦 = βˆ’ 2 π‘₯ βˆ’ 1 2
  • C 𝑦 = βˆ’ π‘₯ 2 + 1
  • D 𝑦 = π‘₯ 2 βˆ’ 3
  • E 𝑦 = π‘₯ 2 + 3

Q20:

Determine the equation of the common tangent to the two curves 𝑦 = 5 π‘₯ βˆ’ π‘₯ βˆ’ 4  and 𝑦 = 9 π‘₯ + 7 π‘₯  .

  • A 1 1 𝑦 + π‘₯ βˆ’ 2 1 = 0
  • B 𝑦 βˆ’ 1 1 π‘₯ βˆ’ 1 3 = 0
  • C 1 1 𝑦 βˆ’ π‘₯ βˆ’ 2 3 = 0
  • D 𝑦 + 1 1 π‘₯ + 9 = 0

Q21:

Find the equations to the tangent lines of the curve 𝑦 = ( π‘₯ + 8 ) ( π‘₯ + 1 0 ) at the points where this curve intersects the π‘₯ -axis.

  • A 𝑦 + 2 π‘₯ + 1 6 = 0 , 𝑦 βˆ’ 2 π‘₯ βˆ’ 2 0 = 0
  • B 𝑦 βˆ’ 2 π‘₯ + 1 6 = 0 , 𝑦 + 2 π‘₯ βˆ’ 2 0 = 0
  • C 𝑦 + 2 π‘₯ βˆ’ 1 6 = 0 , 𝑦 βˆ’ 2 π‘₯ + 2 0 = 0
  • D 𝑦 βˆ’ 2 π‘₯ βˆ’ 1 6 = 0 , 𝑦 + 2 π‘₯ + 2 0 = 0

Q22:

Determine the equation of the tangent to the curve 𝑦 = ο€Ή π‘₯ βˆ’ 8 π‘₯  ο€Ή π‘₯ + 3    at the point ( βˆ’ 1 , 1 8 ) .

  • A 𝑦 + 8 4 π‘₯ + 6 6 = 0
  • B 𝑦 + 4 7 π‘₯ + 2 9 = 0
  • C 𝑦 + 7 π‘₯ βˆ’ 2 5 = 0
  • D 𝑦 βˆ’ 7 π‘₯ βˆ’ 2 5 = 0

Q23:

Find the equation of the tangent to the curve 𝑦 = 5 π‘₯ + 8 π‘₯ + 5 π‘₯ + 6   at the point with π‘₯ -coordinate 0.

  • A π‘₯ = 0
  • B 5 𝑦 + π‘₯ βˆ’ 3 0 = 0
  • C 𝑦 βˆ’ 6 = 0
  • D 𝑦 βˆ’ 5 π‘₯ βˆ’ 6 = 0

Q24:

Find the equation of the tangent to the curve 𝑦 = 6 π‘₯ t a n at π‘₯ = πœ‹ 4 .

  • A βˆ’ 7 𝑦 + 8 4 π‘₯ βˆ’ 2 4 = 0
  • B 4 2 π‘₯ βˆ’ 7 𝑦 βˆ’ 2 4 = 0
  • C 4 2 π‘₯ βˆ’ 1 4 𝑦 βˆ’ 1 8 = 0
  • D 7 𝑦 βˆ’ 8 4 π‘₯ + 2 4 = 0

Q25:

Find the equation of the normal to the curve 𝑦 = π‘₯ t a n at π‘₯ = πœ‹ 4 .

  • A βˆ’ 1 4 π‘₯ + 7 𝑦 βˆ’ 4 = 0
  • B 1 4 π‘₯ βˆ’ 7 𝑦 βˆ’ 4 = 0
  • C 2 8 π‘₯ + 1 4 𝑦 βˆ’ 3 9 = 0
  • D 2 8 𝑦 + 1 4 π‘₯ βˆ’ 3 9 = 0

Nagwa uses cookies to ensure you get the best experience on our website. Learn more about our Privacy Policy.