Worksheet: Tangents to the Graph of a Function

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+8𝑥19 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+5𝑥1=0 touches the curve 𝑓(𝑥)=𝑥𝑥+𝑎. What is 𝑎?

Q9:

If the curve 𝑦=𝑎𝑥+𝑏𝑥+2𝑥+7 is tangent to the line 𝑦=7𝑥3 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 𝑥𝑦3=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=6𝑥4𝑥+20 have a common tangent at the point of intersection? If so, give the equation of this tangent.

  • ANo
  • BYes, 𝑦8𝑥26=0
  • CYes, 𝑦+4𝑥14=0
  • DYes, 𝑦4𝑥22=0

Q12:

Determine the equation of the line tangent to the curve 𝑦=4𝑥2𝑥+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=𝑥1 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+𝑏𝑥4𝑥+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=𝑥2𝑥 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+5𝑥6 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 𝑦4𝑥=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=5𝑥𝑥4 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=𝑥8𝑥𝑥+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

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