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Worksheet: Tangents to the Graph of a Function

Q1:

Find the equation of the tangent to the curve 𝑦 = βˆ’ 2 π‘₯ + 8 π‘₯ βˆ’ 1 9 3 2 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:

Find the equation of the tangent to the curve 𝑦 = 4 π‘₯ βˆ’ 6 π‘₯ + 1 5 3 2 at π‘₯ = βˆ’ 1 .

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

Q3:

Find the equation of the tangent to the curve 𝑦 = βˆ’ 2 π‘₯ βˆ’ 8 π‘₯ + 1 7 3 2 at π‘₯ = βˆ’ 1 .

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

Q4:

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 ) ?

Q5:

If the line 𝑦 = 7 π‘₯ βˆ’ 7 is tangent to the graph of the function 𝑓 at ( 1 , 0 ) , what is 𝑓 β€² ( 1 ) ?

Q6:

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

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

Q7:

What is the π‘₯ -coordinate of the point where the tangent line to 𝑦 = π‘₯ + 9 π‘₯ + 1 2 2 is parallel to the π‘₯ -axis?

  • A βˆ’ 9
  • B 9 2
  • C0
  • D βˆ’ 9 2

Q8:

The point lies on the curve . If the gradient of the tangent there is , what are the values of constants and ?

  • A ,
  • B ,
  • C ,
  • D ,

Q9:

The point lies on the curve . If the gradient of the tangent there is , what are the values of constants and ?

  • A ,
  • B ,
  • C ,
  • D ,

Q10:

Find the equation of the tangent to the curve 𝑦 = π‘₯ + 9 π‘₯ + 2 6 π‘₯ 3 2 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

Q11:

Suppose the line 𝑦 + 5 π‘₯ βˆ’ 1 = 0 touches the curve 𝑓 ( π‘₯ ) = π‘₯ βˆ’ π‘₯ + π‘Ž 2 . What is π‘Ž ?

Q12:

If the curve 𝑦 = π‘Ž π‘₯ + 𝑏 π‘₯ + 2 π‘₯ + 7 3 2 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

Q13:

Find the point on the curve at which the tangent to the curve is parallel to the -axis.

  • A
  • B
  • C
  • D

Q14:

The line π‘₯ βˆ’ 𝑦 βˆ’ 3 = 0 touches the curve 𝑦 = π‘Ž π‘₯ + 𝑏 π‘₯ 3 2 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

Q15:

Do the curves 𝑦 = βˆ’ 2 π‘₯ + 4 π‘₯ + 2 4 2 and 𝑦 = βˆ’ 6 π‘₯ βˆ’ 4 π‘₯ + 2 0 2 meet at a common tangent? 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

Q16:

Determine the equation of the line tangent to the curve 𝑦 = 4 π‘₯ βˆ’ 2 π‘₯ + 4 3 2 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

Q17:

The line 𝑦 + 2 π‘₯ + π‘Ž = 0 is tangent to the curve 𝑦 = π‘₯ βˆ’ 1 2 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

Q18:

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 ) ?

Q19:

The line 5 π‘₯ + 𝑦 = 2 2 touches the curve 𝑦 = π‘Ž π‘₯ + 𝑏 π‘₯ βˆ’ 4 π‘₯ + 2 3 3 2 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

Q20:

Find the equation of the tangent to the curve 𝑦 = π‘₯ βˆ’ 2 π‘₯ 2 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

Q21:

Find the gradient of the tangent to the curve when , approximated to the nearest hundredth, if needed.

  • A
  • B
  • C18.94
  • D

Q22:

Find the equation of the tangent to the curve 𝑦 = 2 + 2 π‘₯ βˆ’ 2 π‘₯ at the point ο€Ό 1 , 9 4  .

  • A 3 π‘₯ 2 2 βˆ’ 𝑦 + 1 3 4 = 0 l n
  • B 3 π‘₯ 2 2 + 𝑦 βˆ’ 1 3 4 = 0 l n
  • C 3 π‘₯ 2 2 + 𝑦 βˆ’ 5 4 = 0 l n
  • D 3 π‘₯ 2 2 βˆ’ 𝑦 βˆ’ 3 2 2 + 9 4 = 0 l n l n

Q23:

Find the equation of the tangent to the curve 𝑦 = 2 + 2 3 π‘₯ βˆ’ π‘₯ at the point ο€Ό 1 , 1 7 2  .

  • A 4 7 π‘₯ 2 2 βˆ’ 𝑦 + 1 9 2 = 0 l n
  • B 4 7 π‘₯ 2 2 + 𝑦 βˆ’ 1 9 2 = 0 l n
  • C 4 7 π‘₯ 2 2 + 𝑦 βˆ’ 1 5 2 = 0 l n
  • D 4 7 π‘₯ 2 2 βˆ’ 𝑦 βˆ’ 4 7 2 2 + 1 7 2 = 0 l n l n

Q24:

List the equations of all the tangents to 𝑦 = βˆ’ π‘₯ 2 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

Q25:

List the equations of all the tangents to 𝑦 = βˆ’ π‘₯ 2 that also lie on the point ( 3 , βˆ’ 8 ) .

  • A 𝑦 βˆ’ 8 π‘₯ βˆ’ 1 6 = 0 , 𝑦 βˆ’ 4 π‘₯ βˆ’ 4 = 0
  • B 𝑦 βˆ’ 8 π‘₯ + 4 8 = 0 , 𝑦 βˆ’ 4 π‘₯ + 1 2 = 0
  • C 𝑦 + 8 π‘₯ + 4 8 = 0 , 𝑦 + 4 π‘₯ + 1 2 = 0
  • D 𝑦 + 8 π‘₯ βˆ’ 1 6 = 0 , 𝑦 + 4 π‘₯ βˆ’ 4 = 0