Worksheet: Intersection Points of Two Functions

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In this worksheet, we will practice using a graphical or algebraic approach to solve systems of equations where one or both are nonlinear functions to find the intersection point of two functions.

Q1:

Find the set of points of intersection of the graphs of 𝑦=3𝑥 and 𝑥+𝑦=40.

  • A{(4,12),(−4,−12)}
  • B{(2,6),(−2,−6)}
  • C{(4,−12),(−12,4)}
  • D{(2,−6),(−6,2)}

Q2:

Answer the following questions for the functions 𝑦=𝑥−3𝑥−4 and 𝑦=𝑥+1.

Complete the table of values for 𝑦=𝑥−3𝑥−4.

𝑥−2−1012
𝑦
  • A4,−2,−3,−7,−8
  • B6,0,−4,−6,−6
  • C4,−2,−4,−6,−6
  • D4,−2,−3,−6,−6
  • E6,0,−3,−7,−8

Complete the table of values for 𝑦=𝑥+1.

𝑥−2−1012
𝑦
  • A−1,0,1,2,3
  • B−2,−1,0,1,2
  • C−3,−2,1,0,1
  • D−3,−2,−1,0,1
  • E−1,0,−1,2,3

Use the tables of values to determine an intersection point of the two graphs.

  • A(−1,−2)
  • B(−2,0)
  • C(−1,0)
  • D(0,−1)
  • E(−2,−1)

By extending the table up to 𝑥=8, check if there are other intersection points. If so, find their coordinates.

  • Ayes, (6,5)
  • Byes, (5,6)
  • Cyes, (4,3)
  • Dyes, (3,4)
  • Eno

Q3:

Answer the following questions.

Determine the equation of the line that passes through the points (2, 0) and (1, 2).

  • A𝑦=−2𝑥+4
  • B𝑦=2𝑥−4
  • C𝑦=−2𝑥−4
  • D𝑦=𝑥+1
  • E𝑦=−𝑥+2

Determine the equation of the line that passes through the points (−1,0) and (0,−2).

  • A𝑦=−2𝑥−4
  • B𝑦=2𝑥−4
  • C𝑦=−2𝑥−2
  • D𝑦=2𝑥+2
  • E𝑦=−2𝑥+4

Hence, do the two lines intersect? If yes, state the point of intersection.

  • AYes, they intersect at (2, 0).
  • BYes, they intersect at (1, 2).
  • CYes, they intersect at (−1,0).
  • DNo, they do not intersect.
  • EYes, they intersect at (0,−2).

Q4:

Answer the following questions.

Determine the equation of the line that passes through the points (−2,−1) and (0,3).

  • A𝑦=−2𝑥+3
  • B𝑦=𝑥+1
  • C𝑦=4𝑥+7
  • D𝑦=2𝑥+3
  • E𝑦=𝑥+3

Determine the equation of the line that passes through the points (−2,4) and (−1,1).

  • A𝑦=−3𝑥−2
  • B𝑦=3𝑥+2
  • C𝑦=−3𝑥+2
  • D𝑦=−3𝑥−4
  • E𝑦=3𝑥+4

Hence, do the two lines intersect? If yes, state the point of intersection.

  • AYes, they intersect at (−2,−1).
  • BYes, they intersect at (−1,1).
  • CNo, they do not intersect.
  • DYes, they intersect at (0,3).
  • EYes, they intersect at (−2,4).

Q5:

Answer the following questions.

Determine the equation of the line that passes through the points (3,1) and (5,3).

  • A𝑦=2𝑥−4
  • B𝑦=−𝑥+2
  • C𝑦=2𝑥−8
  • D𝑦=2𝑥−2
  • E𝑦=𝑥−2

Determine the equation of the line that passes through the points (2,4) and (3,3).

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

Hence, do the two lines intersect? If yes, state the point of intersection.

  • AYes, they intersect at (2,4).
  • BNo, they do not intersect.
  • CYes, they intersect at (4,2).
  • DYes, they intersect at (3,5).
  • EYes, they intersect at (5,3).

Q6:

The given figure shows the graphs of the functions 𝑓(𝑥)=2𝑥−2 and 𝑔(𝑥)=(𝑥)ln. What are the points where 𝑓(𝑥)=𝑔(𝑥)?

  • A(0,1), (−1.594,0.203)
  • B(1,0), (−1.594,0.203)
  • C(0,1), (0.203,−1.549)
  • D(0,−1), (0.203,−1.594)
  • E(1,0), (0.203,−1.594)

Q7:

The given figure shows the graphs of the functions 𝑓(𝑥)=6 and 𝑔(𝑥)=3𝑥−3𝑥. What are the points where 𝑓(𝑥)=𝑔(𝑥)?

  • A(6,−1), (6,2)
  • B(0,1), (0,6)
  • C(6,1), (6,−2)
  • D(1,6), (−2,6)
  • E(−1,6), (2,6)

Q8:

The given figure shows the graphs of the functions 𝑓(𝑥)=3𝑥−3 and 𝑔(𝑥)=𝑥+4𝑥−5. What are the points where 𝑓(𝑥)=𝑔(𝑥)?

  • A(1,0), (−2,−9)
  • B(0,1), (−9,−2)
  • C(1,1), (−2,−9)
  • D(1,1), (−9,−2)
  • E(1,0), (−8,−27)

Q9:

The given figure shows the graphs of the functions 𝑓(𝑥)=4𝑥−2 and 𝑔(𝑥)=−2𝑥+4. What is the point where 𝑓(𝑥)=𝑔(𝑥)?

  • A(1,−2)
  • B(2,1)
  • C(3,−2)
  • D(−2,3)
  • E(1,2)

Q10:

The given figure shows the graphs of the functions 𝑓(𝑥)=5𝑥−4 and 𝑔(𝑥)=−𝑥+8. What is the point where 𝑓(𝑥)=𝑔(𝑥)?

  • A(5,3)
  • B(2,6)
  • C(3,5)
  • D(1,1)
  • E(6,2)

Q11:

The given diagram shows the graph of 𝑓(𝑥)=𝑥−75𝑥50 together with three lines of the form 𝑦=𝑘−𝑥.

By solving 𝑓(𝑥)=−𝑥, determine the coordinates of points 𝐴 and 𝐵.

  • A(−5,5), (5,5)
  • B(−5,5), (5,−5)
  • C(−5,5), (0,0)
  • D(0,0), (5,5)
  • E(−5,−5), (5,5)

Note that from 𝑓(2)=−7125, we know that line 𝑦=2−7125−𝑥 meets 𝑦=𝑓(𝑥) in the point 2,−7125, so (𝑥−2) is a factor of the cubic polynomial 𝑓(𝑥)−2−7125−𝑥. Find the 𝑥-coordinates of the other two intersections.

  • A−√22+1, √22+1
  • B−√22−1, √22−1
  • C−√22+1, √22−1
  • D−√22−1, −√22+1
  • E−√22, √22

Simplify the expression that states that the average rate of change of 𝑓 from 𝑥=𝑐 to 𝑥=𝑑 is −1 to a quadratic equation in 𝑑 that has coefficients involving 𝑐.

  • A𝑑+𝑐𝑑+𝑐=0
  • B4𝑑+2𝑐𝑑+𝑐−25=0
  • C4𝑑+2𝑐𝑑+𝑐−25=0
  • D𝑑+𝑐𝑑+𝑐−25=0
  • E𝑑−𝑐𝑑+𝑐−25=0

The 𝑥-coordinate 𝑐 of point 𝐶 is one where there is just one 𝑑 for which the average rate of change of 𝑓 is −1. By finding the discriminant of the quadratic expression find 𝑐 and 𝑑.

  • A𝑐=10√33, 𝑑=−5√33
  • B𝑐=−10√3, 𝑑=5√3
  • C𝑐=√3, 𝑑=−√32
  • D𝑐=10√3, 𝑑=−5√3
  • E𝑐=10√33, 𝑑=5√33

For what values of 𝑐 are there no points 𝑑 so that the average rate of change of 𝑓 from 𝑥=𝑐 to 𝑥=𝑑 is −1?

  • A𝑐>√33 or 𝑐<−√33
  • B𝑐<10√33 or 𝑐>−10√33
  • C𝑐>10√3 or 𝑐<−10√3
  • D𝑐<√33 or 𝑐>−√33
  • E𝑐>10√33 or 𝑐<−10√33

Q12:

Find all values of 𝑥 where 𝑓(𝑥)=𝑡(𝑥), given 𝑓(𝑥)=(𝑥+34) and 𝑡(𝑥)=𝑥+34.

  • A𝑥=−33 or 𝑥=33
  • B𝑥=−33 or 𝑥=35
  • C𝑥=−34 or 𝑥=35
  • D𝑥=−33 or 𝑥=−34
  • E𝑥=−34 or 𝑥=33

Q13:

Find all the possible values of 𝑥 satisfying 𝑓(𝑥)=𝑥+9𝑥+2 and 𝑓(𝑥)=−16 given 𝑥∈ℤ.

  • A3 or 6
  • B−3
  • C−6
  • D−3 or −6

Q14:

The graphs of 𝑓 and 𝑓 intersect at the point (𝑘,2), where 𝑓(𝑥)=2 and 𝑓(𝑥)=3−𝑥. Find the set of possible values of 𝑘.

  • A{1}
  • B{−1}
  • C{2}
  • D{−2}

Q15:

The given figure shows the graphs of the functions 𝑓(𝑥)=|2𝑥|−5 and 𝑔(𝑥)=(𝑥)ln. What are the points where 𝑓(𝑥)=𝑔(𝑥)?

  • A(0.007,−4.986), (3.059,1.118)
  • B(−4.986,0.007), (1.118,3.059)
  • C(−0.007,−4.986), (3.059,1.118)
  • D(−4.986,0.007), (3.059,1.118)
  • E(0.007,−4.986), (1.118,3.059)

Q16:

Find the 𝑥-intercepts of the function 𝑓(𝑥)=3𝑥−15𝑥−42.

  • A2 and −7
  • B2 and 7
  • C−2 and −7
  • D−2 and 7

Q17:

Find the set of points of intersection of the graphs of 𝑥=𝑦 and 𝑥+𝑦=32.

  • A{(16,16),(−16,−16)}
  • B{(4,4),(−4,−4)}
  • C{(16,−16),(−16,16)}
  • D{(4,−4),(−4,4)}

Q18:

Use technology to plot the graphs of 𝑓(𝑥)=1𝑥+3 and 𝑔(𝑥)=6𝑥+8. Find the coordinates where 𝑓(𝑥)=𝑔(𝑥) if the curves intersect, giving your answer to two decimal places.

  • A(−1.10,0.53), (−0.17,0.35)
  • B(−1.45,−0.69), (0.12,8.69)
  • C(−3.10,−10.57), (−1.24,0.57)
  • D(−3.05,−18.43), (−0.70,0.43)
  • EThe curves do not intersect.

Q19:

Find the coordinates of the point at which the function 𝑓(𝑥)=12 intersects either the 𝑥- or 𝑦-axis.

  • A(12,12)
  • B(12,0)
  • C(0,0)
  • D(0,12)

Q20:

Find the values of 𝑎 and 𝑏 given the function 𝑓(𝑥)=−14𝑥+𝑎 intersects the 𝑦-axis at the point (𝑏,−3).

  • A𝑎=0, 𝑏=3
  • B𝑎=0, 𝑏=−3
  • C𝑎=−3, 𝑏=0
  • D𝑎=3, 𝑏=0

Q21:

If the function 𝑓∶ℕ→ℤ, where 𝑓(𝑥)=(𝑥+17), and the function 𝑔∶ℕ→ℤ, where 𝑔(𝑥)=𝑥+17, find the solution set of 𝑥 which makes 𝑓(𝑥)=𝑔(𝑥).

  • A{−16}
  • B{−17}
  • C{17,16}
  • D∅

Q22:

What is the 𝑥-coordinate of the point where the graphs of 𝑓(𝑥)=9 and 𝑔(𝑥)=6 intersect?

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