Worksheet: Tangents to the Graph of a Function

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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=−2𝑥+8𝑥−19 at 𝑥=2.

  • A − 8 𝑦 + 𝑥 − 2 = 0
  • B 𝑦 − 8 𝑥 + 1 9 = 0
  • C 𝑦 + 8 𝑥 + 1 9 = 0
  • D 𝑦 − 4 𝑥 − 5 = 0

Q2:

If the line 𝑦=3𝑥+9 is tangent to the graph of the function 𝑓 at (2,15), what is 𝑓′(2)?

Q3:

What is the 𝑥-coordinate of the point where the tangent line to 𝑦=𝑥+12𝑥+11 is parallel to the 𝑥-axis?

  • A − 1 2
  • B0
  • C6
  • D − 6

Q4:

The point (3,3) lies on the curve 𝑦=7𝑥+𝑎𝑥+𝑏. If the slope of the tangent there is −1, what are the values of constants 𝑎 and 𝑏?

  • A 𝑎 = − 4 1 , 𝑏 = 6 3
  • B 𝑎 = − 4 3 , 𝑏 = − 1 7
  • C 𝑎 = 4 1 , 𝑏 = − 1 8 3
  • D 𝑎 = − 4 3 , 𝑏 = 6 9

Q5:

Find the point on the curve 𝑦=−40𝑥+40 at which the tangent to the curve is parallel to the 𝑥-axis.

  • A ( 0 , 0 )
  • B ( − 1 , 0 )
  • C ( 1 , 0 )
  • D ( 0 , 4 0 )

Q6:

List the equations of all the tangents to 𝑦=−𝑥 that also lie on the point (2,−3).

  • A 𝑦 − 6 𝑥 + 2 7 = 0 , 𝑦 − 2 𝑥 + 3 = 0
  • B 𝑦 + 6 𝑥 + 2 7 = 0 , 𝑦 + 2 𝑥 + 3 = 0
  • C 𝑦 + 6 𝑥 − 9 = 0 , 𝑦 + 2 𝑥 − 1 = 0
  • D 𝑦 − 6 𝑥 − 9 = 0 , 𝑦 − 2 𝑥 − 1 = 0

Q7:

Find the equation of the tangent to the curve 𝑦=𝑥+9𝑥+26𝑥 that makes an angle of 135∘ with the positive 𝑥-axis.

  • A 𝑦 − 𝑥 3 + 2 3 = 0
  • B 𝑦 + 2 7 𝑥 + 1 0 5 = 0
  • C 𝑦 + 𝑥 + 2 7 = 0
  • D 𝑦 − 8 𝑥 = 0

Q8:

Suppose the line 𝑦+5𝑥−1=0 touches the curve 𝑓(𝑥)=𝑥−𝑥+𝑎. What is 𝑎?

Q9:

If the curve 𝑦=𝑎𝑥+𝑏𝑥+2𝑥+7 is tangent to the line 𝑦=7𝑥−3 at (−1,−10), find the constants 𝑎 and 𝑏.

  • A 𝑎 = − 2 5 , 𝑏 = − 4 0
  • B 𝑎 = − 2 5 , 𝑏 = 4 0
  • C 𝑎 = 5 , 𝑏 = 1 0
  • D 𝑎 = − 4 0 , 𝑏 = − 2 5

Q10:

The line 𝑥−𝑦−3=0 touches the curve 𝑦=𝑎𝑥+𝑏𝑥 at (1,−2). Find 𝑎 and 𝑏.

  • A 𝑎 = − 1 3 , 𝑏 = − 7
  • B 𝑎 = 5 , 𝑏 = − 7
  • C 𝑎 = − 7 , 𝑏 = 5
  • D 𝑎 = 1 3 , 𝑏 = − 1 5

Q11:

Do the curves 𝑦=−2𝑥+4𝑥+24 and 𝑦=−6𝑥−4𝑥+20 have a common tangent at the point of intersection? If so, give the equation of this tangent.

  • ANo
  • BYes, 𝑦−8𝑥−26=0
  • CYes, 𝑦+4𝑥−14=0
  • DYes, 𝑦−4𝑥−22=0

Q12:

Determine the equation of the line tangent to the curve 𝑦=4𝑥−2𝑥+4 at point (−1,−2).

  • A 𝑦 = 8 𝑥 + 6
  • B 𝑦 = 1 6 𝑥 + 1 6
  • C 𝑦 = 1 4 𝑥 + 1 2
  • D 𝑦 = 1 6 𝑥 + 1 4
  • E 𝑦 = 1 6 𝑥 − 2

Q13:

The line 𝑦+2𝑥+𝑎=0 is tangent to the curve 𝑦=𝑥−1 at the point (𝑏,𝑐). Find 𝑎, 𝑏, and 𝑐.

  • A 𝑎 = 4 , 𝑏 = − 2 , 𝑐 = 3
  • B 𝑎 = 2 , 𝑏 = − 1 , 𝑐 = 0
  • C 𝑎 = − 2 , 𝑏 = 1 , 𝑐 = 0
  • D 𝑎 = − 4 , 𝑏 = 2 , 𝑐 = 3

Q14:

The line 𝑦=5𝑥+4 is tangent to the graph of function 𝑓 at the point (−1,−1). What is 𝑓′(−1)?

Q15:

The line 5𝑥+𝑦=22 touches the curve 𝑦=𝑎𝑥+𝑏𝑥−4𝑥+23 at the point (1,17). Find 𝑎 and 𝑏.

  • A 𝑎 = 3 , 𝑏 = − 5
  • B 𝑎 = − 5 , 𝑏 = 3
  • C 𝑎 = − 7 , 𝑏 = 5
  • D 𝑎 = − 7 , 𝑏 = − 5

Q16:

Find the equation of the tangent to the curve 𝑦=𝑥−2𝑥 at the point (𝑥,3) on the curve.

  • A 𝑦 − 4 𝑥 + 9 = 0 , 𝑦 + 4 𝑥 + 1 = 0
  • B 4 𝑦 − 𝑥 − 9 = 0 , − 4 𝑦 − 𝑥 + 1 1 = 0
  • C − 4 𝑦 − 𝑥 + 1 5 = 0 , 4 𝑦 − 𝑥 − 1 3 = 0
  • D 𝑦 + 4 𝑥 − 1 5 = 0 , 𝑦 − 4 𝑥 − 7 = 0

Q17:

Find the equation of the tangent to the curve 𝑦=8𝑥+5𝑥−6 at the point (−1,−3).

  • A 𝑦 − 1 1 𝑥 − 8 = 0
  • B 1 1 𝑦 − 𝑥 + 3 2 = 0
  • C − 1 1 𝑦 − 𝑥 − 3 4 = 0
  • D 𝑦 + 1 1 𝑥 + 1 4 = 0

Q18:

List the equations of the normals to 𝑦=𝑥+2𝑥 at the points where this curve meets the line 𝑦−4𝑥=0.

  • A 2 𝑦 + 𝑥 = 0 , 6 𝑦 + 𝑥 − 5 0 = 0
  • B 𝑦 + 2 𝑥 = 0 , 𝑦 + 6 𝑥 − 4 = 0
  • C 𝑦 − 2 𝑥 = 0 , 𝑦 − 6 𝑥 + 4 = 0
  • D 2 𝑦 − 𝑥 = 0 , 6 𝑦 − 𝑥 − 4 6 = 0

Q19:

Determine the equation of the line normal to the curve 4𝑦=𝑥 at the point (2,−2).

  • A 𝑦 = − 2 𝑥 − 1 2
  • B 𝑦 = 𝑥 2 + 3
  • C 𝑦 = − 𝑥 2 + 1
  • D 𝑦 = 𝑥 2 − 3
  • E 𝑦 = − 𝑥 2 − 1

Q20:

Determine the equation of the common tangent to the two curves 𝑦=5𝑥−𝑥−4 and 𝑦=9𝑥+7𝑥.

  • A 𝑦 − 1 1 𝑥 − 1 3 = 0
  • B 𝑦 + 1 1 𝑥 + 9 = 0
  • C 1 1 𝑦 − 𝑥 − 2 3 = 0
  • D 1 1 𝑦 + 𝑥 − 2 1 = 0

Q21:

Find the equations to the tangent lines of the curve 𝑦=(𝑥+8)(𝑥+10) at the points where this curve intersects the 𝑥-axis.

  • A 𝑦 + 2 𝑥 + 1 6 = 0 , 𝑦 − 2 𝑥 − 2 0 = 0
  • B 𝑦 − 2 𝑥 − 1 6 = 0 , 𝑦 + 2 𝑥 + 2 0 = 0
  • C 𝑦 + 2 𝑥 − 1 6 = 0 , 𝑦 − 2 𝑥 + 2 0 = 0
  • D 𝑦 − 2 𝑥 + 1 6 = 0 , 𝑦 + 2 𝑥 − 2 0 = 0

Q22:

Determine the equation of the tangent to the curve 𝑦=𝑥−8𝑥𝑥+3 at the point (−1,18).

  • A 𝑦 + 4 7 𝑥 + 2 9 = 0
  • B 𝑦 + 8 4 𝑥 + 6 6 = 0
  • C 𝑦 + 7 𝑥 − 2 5 = 0
  • D 𝑦 − 7 𝑥 − 2 5 = 0

Q23:

Find the equation of the tangent to the curve 𝑦=5𝑥+8𝑥+5𝑥+6 at the point with 𝑥-coordinate 0.

  • A 𝑦 − 5 𝑥 − 6 = 0
  • B 𝑦 − 6 = 0
  • C 5 𝑦 + 𝑥 − 3 0 = 0
  • D 𝑥 = 0

Q24:

Find the equation of the tangent to the curve 𝑦=6𝑥tan at 𝑥=𝜋4.

  • A 7 𝑦 − 8 4 𝑥 + 2 4 = 0
  • B 4 2 𝑥 − 7 𝑦 − 2 4 = 0
  • C − 7 𝑦 + 8 4 𝑥 − 2 4 = 0
  • D 4 2 𝑥 − 1 4 𝑦 − 1 8 = 0

Q25:

Find the equation of the normal to the curve 𝑦=𝑥tan at 𝑥=𝜋4.

  • A − 1 4 𝑥 + 7 𝑦 − 4 = 0
  • B 2 8 𝑦 + 1 4 𝑥 − 3 9 = 0
  • C 2 8 𝑥 + 1 4 𝑦 − 3 9 = 0
  • D 1 4 𝑥 − 7 𝑦 − 4 = 0

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