Worksheet: Intersection Points of Two Functions

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=3π‘₯ and π‘₯+𝑦=40.

  • A { ( 4 , 1 2 ) , ( βˆ’ 4 , βˆ’ 1 2 ) }
  • B { ( 2 , 6 ) , ( βˆ’ 2 , βˆ’ 6 ) }
  • C { ( 4 , βˆ’ 1 2 ) , ( βˆ’ 1 2 , 4 ) }
  • D { ( 2 , βˆ’ 6 ) , ( βˆ’ 6 , 2 ) }

Q2:

Answer the following questions for the functions 𝑦=π‘₯βˆ’3π‘₯βˆ’4 and 𝑦=π‘₯+1.

Complete the table of values for 𝑦=π‘₯βˆ’3π‘₯βˆ’4.

π‘₯ βˆ’ 2 βˆ’ 1 0 1 2
𝑦
  • A 4 , βˆ’ 2 , βˆ’ 3 , βˆ’ 7 , βˆ’ 8
  • B 6 , 0 , βˆ’ 4 , βˆ’ 6 , βˆ’ 6
  • C 4 , βˆ’ 2 , βˆ’ 4 , βˆ’ 6 , βˆ’ 6
  • D 4 , βˆ’ 2 , βˆ’ 3 , βˆ’ 6 , βˆ’ 6
  • E 6 , 0 , βˆ’ 3 , βˆ’ 7 , βˆ’ 8

Complete the table of values for 𝑦=π‘₯+1.

π‘₯ βˆ’ 2 βˆ’ 1 0 1 2
𝑦
  • A βˆ’ 1 , 0 , 1 , 2 , 3
  • B βˆ’ 2 , βˆ’ 1 , 0 , 1 , 2
  • C βˆ’ 3 , βˆ’ 2 , 1 , 0 , 1
  • D βˆ’ 3 , βˆ’ 2 , βˆ’ 1 , 0 , 1
  • E βˆ’ 1 , 0 , βˆ’ 1 , 2 , 3

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

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

By extending the table up to π‘₯=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).

  • A 𝑦 = βˆ’ 2 π‘₯ + 4
  • B 𝑦 = 2 π‘₯ βˆ’ 4
  • C 𝑦 = βˆ’ 2 π‘₯ βˆ’ 4
  • D 𝑦 = π‘₯ + 1
  • E 𝑦 = βˆ’ π‘₯ + 2

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

  • A 𝑦 = βˆ’ 2 π‘₯ βˆ’ 4
  • B 𝑦 = 2 π‘₯ βˆ’ 4
  • C 𝑦 = βˆ’ 2 π‘₯ βˆ’ 2
  • D 𝑦 = 2 π‘₯ + 2
  • E 𝑦 = βˆ’ 2 π‘₯ + 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 (βˆ’1,0).
  • DNo, they do not intersect.
  • EYes, they intersect at (0,βˆ’2).

Q4:

Answer the following questions.

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

  • A 𝑦 = βˆ’ 2 π‘₯ + 3
  • B 𝑦 = π‘₯ + 1
  • C 𝑦 = 4 π‘₯ + 7
  • D 𝑦 = 2 π‘₯ + 3
  • E 𝑦 = π‘₯ + 3

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

  • A 𝑦 = βˆ’ 3 π‘₯ βˆ’ 2
  • B 𝑦 = 3 π‘₯ + 2
  • C 𝑦 = βˆ’ 3 π‘₯ + 2
  • D 𝑦 = βˆ’ 3 π‘₯ βˆ’ 4
  • E 𝑦 = 3 π‘₯ + 4

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

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

Q5:

Answer the following questions.

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

  • A 𝑦 = 2 π‘₯ βˆ’ 4
  • B 𝑦 = βˆ’ π‘₯ + 2
  • C 𝑦 = 2 π‘₯ βˆ’ 8
  • D 𝑦 = 2 π‘₯ βˆ’ 2
  • E 𝑦 = π‘₯ βˆ’ 2

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

  • A 𝑦 = βˆ’ π‘₯ + 2
  • B 𝑦 = βˆ’ π‘₯ + 3
  • C 𝑦 = π‘₯ + 6
  • D 𝑦 = βˆ’ π‘₯ + 6
  • E 𝑦 = π‘₯ + 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(π‘₯)=2π‘₯βˆ’2 and 𝑔(π‘₯)=(π‘₯)ln. What are the points where 𝑓(π‘₯)=𝑔(π‘₯)?

  • A ( 0 , 1 ) , ( βˆ’ 1 . 5 9 4 , 0 . 2 0 3 )
  • B ( 1 , 0 ) , ( βˆ’ 1 . 5 9 4 , 0 . 2 0 3 )
  • C ( 0 , 1 ) , ( 0 . 2 0 3 , βˆ’ 1 . 5 4 9 )
  • D ( 0 , βˆ’ 1 ) , ( 0 . 2 0 3 , βˆ’ 1 . 5 9 4 )
  • E ( 1 , 0 ) , ( 0 . 2 0 3 , βˆ’ 1 . 5 9 4 )

Q7:

The given figure shows the graphs of the functions 𝑓(π‘₯)=6 and 𝑔(π‘₯)=3π‘₯βˆ’3π‘₯. What are the points where 𝑓(π‘₯)=𝑔(π‘₯)?

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

Q8:

The given figure shows the graphs of the functions 𝑓(π‘₯)=3π‘₯βˆ’3 and 𝑔(π‘₯)=π‘₯+4π‘₯βˆ’5. What are the points where 𝑓(π‘₯)=𝑔(π‘₯)?

  • A ( 1 , 0 ) , ( βˆ’ 2 , βˆ’ 9 )
  • B ( 0 , 1 ) , ( βˆ’ 9 , βˆ’ 2 )
  • C ( 1 , 1 ) , ( βˆ’ 2 , βˆ’ 9 )
  • D ( 1 , 1 ) , ( βˆ’ 9 , βˆ’ 2 )
  • E ( 1 , 0 ) , ( βˆ’ 8 , βˆ’ 2 7 )

Q9:

The given figure shows the graphs of the functions 𝑓(π‘₯)=4π‘₯βˆ’2 and 𝑔(π‘₯)=βˆ’2π‘₯+4. What is the point where 𝑓(π‘₯)=𝑔(π‘₯)?

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

Q10:

The given figure shows the graphs of the functions 𝑓(π‘₯)=5π‘₯βˆ’4 and 𝑔(π‘₯)=βˆ’π‘₯+8. What is the point where 𝑓(π‘₯)=𝑔(π‘₯)?

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

Q11:

The given diagram shows the graph of 𝑓(π‘₯)=ο€Ήπ‘₯βˆ’75π‘₯50 together with three lines of the form 𝑦=π‘˜βˆ’π‘₯.

By solving 𝑓(π‘₯)=βˆ’π‘₯, determine the coordinates of points 𝐴 and 𝐡.

  • A ( βˆ’ 5 , 5 ) , ( 5 , 5 )
  • B ( βˆ’ 5 , 5 ) , ( 5 , βˆ’ 5 )
  • C ( βˆ’ 5 , 5 ) , ( 0 , 0 )
  • D ( 0 , 0 ) , ( 5 , 5 )
  • E ( βˆ’ 5 , βˆ’ 5 ) , ( 5 , 5 )

Note that from 𝑓(2)=βˆ’7125, we know that line 𝑦=2βˆ’7125βˆ’π‘₯ meets 𝑦=𝑓(π‘₯) in the point ο€Ό2,βˆ’7125, so (π‘₯βˆ’2) is a factor of the cubic polynomial 𝑓(π‘₯)βˆ’ο€Ό2βˆ’7125βˆ’π‘₯. Find the π‘₯-coordinates of the other two intersections.

  • A βˆ’ √ 2 2 + 1 , √ 2 2 + 1
  • B βˆ’ √ 2 2 βˆ’ 1 , √ 2 2 βˆ’ 1
  • C βˆ’ √ 2 2 + 1 , √ 2 2 βˆ’ 1
  • D βˆ’ √ 2 2 βˆ’ 1 , βˆ’ √ 2 2 + 1
  • E βˆ’ √ 2 2 , √ 2 2

Simplify the expression that states that the average rate of change of 𝑓 from π‘₯=𝑐 to π‘₯=𝑑 is βˆ’1 to a quadratic equation in 𝑑 that has coefficients involving 𝑐.

  • A 𝑑 + 𝑐 𝑑 + 𝑐 = 0  
  • B 4 𝑑 + 2 𝑐 𝑑 + ο€Ή 𝑐 βˆ’ 2 5  = 0  
  • C 4 𝑑 + 2 𝑐 𝑑 + ο€Ή 𝑐 βˆ’ 2 5  = 0  
  • D 𝑑 + 𝑐 𝑑 + ο€Ή 𝑐 βˆ’ 2 5  = 0  
  • E 𝑑 βˆ’ 𝑐 𝑑 + ο€Ή 𝑐 βˆ’ 2 5  = 0  

The π‘₯-coordinate 𝑐 of point 𝐢 is one where there is just one 𝑑 for which the average rate of change of 𝑓 is βˆ’1. By finding the discriminant of the quadratic expression find 𝑐 and 𝑑.

  • A 𝑐 = 1 0 √ 3 3 , 𝑑 = βˆ’ 5 √ 3 3
  • B 𝑐 = βˆ’ 1 0 √ 3 , 𝑑 = 5 √ 3
  • C 𝑐 = √ 3 , 𝑑 = βˆ’ √ 3 2
  • D 𝑐 = 1 0 √ 3 , 𝑑 = βˆ’ 5 √ 3
  • E 𝑐 = 1 0 √ 3 3 , 𝑑 = 5 √ 3 3

For what values of 𝑐 are there no points 𝑑 so that the average rate of change of 𝑓 from π‘₯=𝑐 to π‘₯=𝑑 is βˆ’1?

  • A 𝑐 > √ 3 3 or 𝑐<βˆ’βˆš33
  • B 𝑐 < 1 0 √ 3 3 or 𝑐>βˆ’10√33
  • C 𝑐 > 1 0 √ 3 or 𝑐<βˆ’10√3
  • D 𝑐 < √ 3 3 or 𝑐>βˆ’βˆš33
  • E 𝑐 > 1 0 √ 3 3 or 𝑐<βˆ’10√33

Q12:

Find all values of π‘₯ where 𝑓(π‘₯)=𝑑(π‘₯), given 𝑓(π‘₯)=(π‘₯+34) and 𝑑(π‘₯)=π‘₯+34.

  • A π‘₯ = βˆ’ 3 3 or π‘₯=33
  • B π‘₯ = βˆ’ 3 3 or π‘₯=35
  • C π‘₯ = βˆ’ 3 4 or π‘₯=35
  • D π‘₯ = βˆ’ 3 3 or π‘₯=βˆ’34
  • E π‘₯ = βˆ’ 3 4 or π‘₯=33

Q13:

Find all the possible values of π‘₯ satisfying 𝑓(π‘₯)=π‘₯+9π‘₯+2 and 𝑓(π‘₯)=βˆ’16 given π‘₯βˆˆβ„€.

  • A3 or 6
  • B βˆ’ 3
  • C βˆ’ 6
  • D βˆ’ 3 or βˆ’6

Q14:

The graphs of π‘“οŠ§ and π‘“οŠ¨ intersect at the point (π‘˜,2), where 𝑓(π‘₯)=2οŠ§ο— and 𝑓(π‘₯)=3βˆ’π‘₯. Find the set of possible values of π‘˜.

  • A { 1 }
  • B { βˆ’ 1 }
  • C { 2 }
  • D { βˆ’ 2 }

Q15:

The given figure shows the graphs of the functions 𝑓(π‘₯)=|2π‘₯|βˆ’5 and 𝑔(π‘₯)=(π‘₯)ln. What are the points where 𝑓(π‘₯)=𝑔(π‘₯)?

  • A ( 0 . 0 0 7 , βˆ’ 4 . 9 8 6 ) , ( 3 . 0 5 9 , 1 . 1 1 8 )
  • B ( βˆ’ 4 . 9 8 6 , 0 . 0 0 7 ) , ( 1 . 1 1 8 , 3 . 0 5 9 )
  • C ( βˆ’ 0 . 0 0 7 , βˆ’ 4 . 9 8 6 ) , ( 3 . 0 5 9 , 1 . 1 1 8 )
  • D ( βˆ’ 4 . 9 8 6 , 0 . 0 0 7 ) , ( 3 . 0 5 9 , 1 . 1 1 8 )
  • E ( 0 . 0 0 7 , βˆ’ 4 . 9 8 6 ) , ( 1 . 1 1 8 , 3 . 0 5 9 )

Q16:

Find the π‘₯-intercepts of the function 𝑓(π‘₯)=3π‘₯βˆ’15π‘₯βˆ’42.

  • A2 and βˆ’7
  • B2 and 7
  • C βˆ’ 2 and βˆ’7
  • D βˆ’ 2 and 7

Q17:

Find the set of points of intersection of the graphs of π‘₯=𝑦 and π‘₯+𝑦=32.

  • A { ( 1 6 , 1 6 ) , ( βˆ’ 1 6 , βˆ’ 1 6 ) }
  • B { ( 4 , 4 ) , ( βˆ’ 4 , βˆ’ 4 ) }
  • C { ( 1 6 , βˆ’ 1 6 ) , ( βˆ’ 1 6 , 1 6 ) }
  • D { ( 4 , βˆ’ 4 ) , ( βˆ’ 4 , 4 ) }

Q18:

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

  • A ( βˆ’ 1 . 1 0 , 0 . 5 3 ) , ( βˆ’ 0 . 1 7 , 0 . 3 5 )
  • B ( βˆ’ 1 . 4 5 , βˆ’ 0 . 6 9 ) , ( 0 . 1 2 , 8 . 6 9 )
  • C ( βˆ’ 3 . 1 0 , βˆ’ 1 0 . 5 7 ) , ( βˆ’ 1 . 2 4 , 0 . 5 7 )
  • D ( βˆ’ 3 . 0 5 , βˆ’ 1 8 . 4 3 ) , ( βˆ’ 0 . 7 0 , 0 . 4 3 )
  • EThe curves do not intersect.

Q19:

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

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

Q20:

Find the values of π‘Ž and 𝑏 given the function 𝑓(π‘₯)=βˆ’14π‘₯+π‘Ž intersects the 𝑦-axis at the point (𝑏,βˆ’3).

  • A π‘Ž = 0 , 𝑏 = 3
  • B π‘Ž = 0 , 𝑏 = βˆ’ 3
  • C π‘Ž = βˆ’ 3 , 𝑏 = 0
  • D π‘Ž = 3 , 𝑏 = 0

Q21:

If the function π‘“βˆΆβ„•β†’β„€, where 𝑓(π‘₯)=(π‘₯+17), and the function π‘”βˆΆβ„•β†’β„€, where 𝑔(π‘₯)=π‘₯+17, find the solution set of π‘₯ which makes 𝑓(π‘₯)=𝑔(π‘₯).

  • A { βˆ’ 1 6 }
  • B { βˆ’ 1 7 }
  • C { 1 7 , 1 6 }
  • D βˆ…

Q22:

What is the π‘₯-coordinate of the point where the graphs of 𝑓(π‘₯)=9ο—οŠ±οŠ¨οŠ¦ and 𝑔(π‘₯)=6ο—οŠ±οŠ¨οŠ¦ intersect?

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