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−8𝑦+𝑥−2=0
  • B𝑦−8𝑥+19=0
  • C𝑦+8𝑥+19=0
  • D𝑦−4𝑥−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−12
  • B0
  • C6
  • D−6

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.

  • A𝑦−𝑥3+23=0
  • B𝑦+27𝑥+105=0
  • C𝑦+𝑥+27=0
  • D𝑦−8𝑥=0

Q5:

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.

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

Q6:

The line 𝑥−𝑦−3=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=−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

Q8:

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

  • A𝑦=8𝑥+6
  • B𝑦=16𝑥+16
  • C𝑦=14𝑥+12
  • D𝑦=16𝑥+14
  • E𝑦=16𝑥−2

Q9:

The line 𝑦+2𝑥+𝑎=0 is tangent to the curve 𝑦=𝑥−1 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=𝑥−2𝑥 at the point (𝑥,3) on the curve.

  • A𝑦−4𝑥+9=0, 𝑦+4𝑥+1=0
  • B4𝑦−𝑥−9=0, −4𝑦−𝑥+11=0
  • C−4𝑦−𝑥+15=0, 4𝑦−𝑥−13=0
  • D𝑦+4𝑥−15=0, 𝑦−4𝑥−7=0

Q12:

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

  • A2𝑦+𝑥=0, 6𝑦+𝑥−50=0
  • B𝑦+2𝑥=0, 𝑦+6𝑥−4=0
  • C𝑦−2𝑥=0, 𝑦−6𝑥+4=0
  • D2𝑦−𝑥=0, 6𝑦−𝑥−46=0

Q13:

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

  • A𝑦=−2𝑥−12
  • B𝑦=𝑥2+3
  • C𝑦=−𝑥2+1
  • D𝑦=𝑥2−3
  • E𝑦=−𝑥2−1

Q14:

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

  • A𝑦−11𝑥−13=0
  • B𝑦+11𝑥+9=0
  • C11𝑦−𝑥−23=0
  • D11𝑦+𝑥−21=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, 𝑦−2𝑥−20=0
  • B𝑦−2𝑥−16=0, 𝑦+2𝑥+20=0
  • C𝑦+2𝑥−16=0, 𝑦−2𝑥+20=0
  • D𝑦−2𝑥+16=0, 𝑦+2𝑥−20=0

Q16:

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

  • A−14𝑥+7𝑦−4=0
  • B28𝑦+14𝑥−39=0
  • C28𝑥+14𝑦−39=0
  • D14𝑥−7𝑦−4=0

Q17:

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

  • A−9𝑦−𝑥+8=0, −9𝑦−𝑥−116=0
  • B𝑦−9𝑥+8=0, 𝑦−9𝑥+4=0
  • C𝑦+9𝑥−10=0, 𝑦+9𝑥+22=0
  • D9𝑦−𝑥−10=0, 9𝑦−𝑥+118=0

Q18:

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

  • A−𝑥2+𝑦−12=0
  • B𝑥2+𝑦−32=0
  • C2𝑥+𝑦−3=0
  • D−2𝑥+𝑦+1=0

Q19:

Two tangents to 𝑦=4𝑥−6𝑥+7𝑥−3 are parallel. Given that one tangent is at (1,2), find the equation of the other one.

  • A−7𝑥+𝑦+3=0
  • B7𝑥+𝑦+3=0
  • C𝑥7+𝑦+3=0
  • D−𝑥7+𝑦+3=0

Q20:

Find the equations of the tangents to the curve 𝑦=−𝑥+4𝑥−18 that are parallel to the straight line −𝑥+𝑦=−3.

  • A𝑦−𝑥−16=0, 𝑦−𝑥−20=0
  • B𝑦−𝑥−16=0, 𝑦−𝑥+20=0
  • C𝑦−𝑥+16=0, 𝑦−𝑥−20=0
  • D𝑦−𝑥+16=0, 𝑦−𝑥+20=0

Q21:

Find the equation of the normal to the curve 𝑦=−2𝑥−7𝑥+2 at 𝑥=−2.

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

  • A𝑦−10𝑥+25=0
  • B𝑦+10𝑥−25=0
  • C10𝑦+𝑥−255=0
  • D10𝑦−𝑥−245=0

Q23:

Determine the equation of the tangent to the curve 𝑦=−𝑥−3𝑥 at point (−1,4).

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

Q24:

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

  • A8𝑦+𝑥+123=0
  • B𝑦−8𝑥−9=0
  • C8𝑦−𝑥+117=0
  • D𝑦+4𝑥+9=0
  • E𝑦+8𝑥+9=0

Q25:

Find the equation of the tangent to the curve 𝑦=2𝑥−45𝑥sincos at 𝑥=3𝜋2.

  • A−22𝑦+𝑥−3𝜋2=0
  • B𝑦+22𝑥+33𝜋=0
  • C𝑦+22𝑥−33𝜋=0
  • D𝑦−22𝑥+33𝜋=0

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