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.

  • A8𝑦+𝑥2=0
  • B𝑦8𝑥+19=0
  • C𝑦+8𝑥+19=0
  • D𝑦4𝑥5=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?

  • A12
  • B0
  • C6
  • D6

Q4:

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

  • A𝑦𝑥3+23=0
  • B𝑦+27𝑥+105=0
  • C𝑦+𝑥+27=0
  • D𝑦8𝑥=0

Q5:

If the curve 𝑦=𝑎𝑥+𝑏𝑥+2𝑥+7 is tangent to the line 𝑦=7𝑥3 at (1,10), find the constants 𝑎 and 𝑏.

  • A𝑎=25, 𝑏=40
  • B𝑎=25, 𝑏=40
  • C𝑎=5, 𝑏=10
  • D𝑎=40, 𝑏=25

Q6:

The line 𝑥𝑦3=0 touches the curve 𝑦=𝑎𝑥+𝑏𝑥 at (1,2). Find 𝑎 and 𝑏.

  • A𝑎=13, 𝑏=7
  • B𝑎=5, 𝑏=7
  • C𝑎=7, 𝑏=5
  • D𝑎=13, 𝑏=15

Q7:

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

Q8:

Determine the equation of the line tangent to the curve 𝑦=4𝑥2𝑥+4 at point (1,2).

  • A𝑦=8𝑥+6
  • B𝑦=16𝑥+16
  • C𝑦=14𝑥+12
  • D𝑦=16𝑥+14
  • E𝑦=16𝑥2

Q9:

The line 𝑦+2𝑥+𝑎=0 is tangent to the curve 𝑦=𝑥1 at the point (𝑏,𝑐). Find 𝑎, 𝑏, and 𝑐.

  • A𝑎=4, 𝑏=2, 𝑐=3
  • B𝑎=2, 𝑏=1, 𝑐=0
  • C𝑎=2, 𝑏=1, 𝑐=0
  • D𝑎=4, 𝑏=2, 𝑐=3

Q10:

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

Q11:

Find the equation of the tangent to the curve 𝑦=𝑥2𝑥 at the point (𝑥,3) on the curve.

  • A𝑦4𝑥+9=0, 𝑦+4𝑥+1=0
  • B4𝑦𝑥9=0, 4𝑦𝑥+11=0
  • C4𝑦𝑥+15=0, 4𝑦𝑥13=0
  • D𝑦+4𝑥15=0, 𝑦4𝑥7=0

Q12:

List the equations of the normals to 𝑦=𝑥+2𝑥 at the points where this curve meets the line 𝑦4𝑥=0.

  • A2𝑦+𝑥=0, 6𝑦+𝑥50=0
  • B𝑦+2𝑥=0, 𝑦+6𝑥4=0
  • C𝑦2𝑥=0, 𝑦6𝑥+4=0
  • D2𝑦𝑥=0, 6𝑦𝑥46=0

Q13:

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

  • A𝑦=2𝑥12
  • B𝑦=𝑥2+3
  • C𝑦=𝑥2+1
  • D𝑦=𝑥23
  • E𝑦=𝑥21

Q14:

Determine the equation of the common tangent to the two curves 𝑦=5𝑥𝑥4 and 𝑦=9𝑥+7𝑥.

  • A𝑦11𝑥13=0
  • B𝑦+11𝑥+9=0
  • C11𝑦𝑥23=0
  • D11𝑦+𝑥21=0

Q15:

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.

  • A𝑦+2𝑥+16=0, 𝑦2𝑥20=0
  • B𝑦2𝑥16=0, 𝑦+2𝑥+20=0
  • C𝑦+2𝑥16=0, 𝑦2𝑥+20=0
  • D𝑦2𝑥+16=0, 𝑦+2𝑥20=0

Q16:

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

  • A14𝑥+7𝑦4=0
  • B28𝑦+14𝑥39=0
  • C28𝑥+14𝑦39=0
  • D14𝑥7𝑦4=0

Q17:

Find the equations of the two tangents to the curve 𝑦=𝑥+6𝑥6 that are perpendicular to the line 𝑥+9𝑦=9.

  • A9𝑦𝑥+8=0, 9𝑦𝑥116=0
  • B𝑦9𝑥+8=0, 𝑦9𝑥+4=0
  • C𝑦+9𝑥10=0, 𝑦+9𝑥+22=0
  • D9𝑦𝑥10=0, 9𝑦𝑥+118=0

Q18:

Find the equation of the normal to the curve 𝑦=6𝑥6𝑥+1 at 𝑥=1.

  • A𝑥2+𝑦12=0
  • B𝑥2+𝑦32=0
  • C2𝑥+𝑦3=0
  • D2𝑥+𝑦+1=0

Q19:

Two tangents to 𝑦=4𝑥6𝑥+7𝑥3 are parallel. Given that one tangent is at (1,2), find the equation of the other one.

  • A7𝑥+𝑦+3=0
  • B7𝑥+𝑦+3=0
  • C𝑥7+𝑦+3=0
  • D𝑥7+𝑦+3=0

Q20:

Find the equations of the tangents to the curve 𝑦=𝑥+4𝑥18 that are parallel to the straight line 𝑥+𝑦=3.

  • A𝑦𝑥16=0, 𝑦𝑥20=0
  • B𝑦𝑥16=0, 𝑦𝑥+20=0
  • C𝑦𝑥+16=0, 𝑦𝑥20=0
  • D𝑦𝑥+16=0, 𝑦𝑥+20=0

Q21:

Find the equation of the normal to the curve 𝑦=2𝑥7𝑥+2 at 𝑥=2.

  • A𝑦+2𝑥+6=0
  • B4𝑦+𝑥+42=0
  • C𝑦+4𝑥+2=0
  • D𝑦+6𝑥2=0

Q22:

Find the equation of the tangent to the curve 𝑓(𝑥)=𝑥 at its point of intersection with the curve 𝑔(𝑥)=125𝑥.

  • A𝑦10𝑥+25=0
  • B𝑦+10𝑥25=0
  • C10𝑦+𝑥255=0
  • D10𝑦𝑥245=0

Q23:

Determine the equation of the tangent to the curve 𝑦=𝑥3𝑥 at point (1,4).

  • A𝑦=2𝑥+6
  • B𝑦=2𝑥6
  • C𝑦=3𝑥+7
  • D𝑦=3𝑥+1
  • E𝑦=2𝑥2

Q24:

Find the equation of the tangent to the curve 𝑦=𝑥|𝑥|+2𝑥 at 𝑥=3.

  • A8𝑦+𝑥+123=0
  • B𝑦8𝑥9=0
  • C8𝑦𝑥+117=0
  • D𝑦+4𝑥+9=0
  • E𝑦+8𝑥+9=0

Q25:

Find the equation of the tangent to the curve 𝑦=2𝑥45𝑥sincos at 𝑥=3𝜋2.

  • A22𝑦+𝑥3𝜋2=0
  • B𝑦+22𝑥+33𝜋=0
  • C𝑦+22𝑥33𝜋=0
  • D𝑦22𝑥+33𝜋=0

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