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Lesson: Finding the Intersection Points of Two Functions

In this lesson, we will learn how to find the intersection points of two functions, using a graphical or algebraic approach to solve systems of equations where one or both are nonlinear.

Video

08:50

Sample Question Videos

03:53

Worksheet • 17 Questions • 1 Video

Q1:

Find the set of points of intersection of the graphs of 𝑦 = 3 π‘₯ and π‘₯ + 𝑦 = 4 0 2 2 .

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

Q2:

Find the set of points of intersection of the graphs of π‘₯ = 𝑦 and π‘₯ + 𝑦 = 3 2 2 2 .

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

Q3:

Find the set of points of intersection of the graphs of π‘₯ = 𝑦 and π‘₯ + 𝑦 = 1 6 2 2 2 .

  • A { ( 9 , 9 ) , ( βˆ’ 9 , βˆ’ 9 ) }
  • B { ( 9 , βˆ’ 9 ) , ( βˆ’ 9 , 9 ) }
  • C { ( 8 1 , βˆ’ 8 1 ) , ( βˆ’ 8 1 , 8 1 ) }
  • D { ( 8 1 , 8 1 ) , ( βˆ’ 8 1 , βˆ’ 8 1 ) }

Q4:

Find the set of points of intersection of the graphs of π‘₯ = 𝑦 and π‘₯ + 𝑦 = 2 0 0 2 2 .

  • A { ( 1 0 , 1 0 ) , ( βˆ’ 1 0 , βˆ’ 1 0 ) }
  • B { ( 1 0 , βˆ’ 1 0 ) , ( βˆ’ 1 0 , 1 0 ) }
  • C { ( 1 0 0 , βˆ’ 1 0 0 ) , ( βˆ’ 1 0 0 , 1 0 0 ) }
  • D { ( 1 0 0 , 1 0 0 ) , ( βˆ’ 1 0 0 , βˆ’ 1 0 0 ) }

Q5:

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

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

Q6:

The given figure shows the graphs of the functions 𝑓 ( π‘₯ ) = 6 and 𝑔 ( π‘₯ ) = 3 π‘₯ βˆ’ 3 π‘₯ 2 . What are the points where 𝑓 ( π‘₯ ) = 𝑔 ( π‘₯ ) ?

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

Q7:

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

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

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

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

  • ANo, they do not intersect.
  • BYes, they intersect at (2, 0).
  • CYes, they intersect at (1, 2).
  • DYes, they intersect at ( βˆ’ 1 , 0 ) .
  • EYes, they intersect at ( 0 , βˆ’ 2 ) .

Q8:

The graphs of 𝑓 1 and 𝑓 2 intersect at the point ( π‘˜ , 2 ) , where 𝑓 ( π‘₯ ) = 2 1 π‘₯ and 𝑓 ( π‘₯ ) = 3 βˆ’ π‘₯ 2 . Find the set of possible values of π‘˜ .

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

Q9:

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

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

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

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

  • A ( βˆ’ 1 , 0 )
  • B ( βˆ’ 2 , 0 )
  • C ( 0 , βˆ’ 1 )
  • D ( βˆ’ 1 , βˆ’ 2 )
  • 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, ( 5 , 6 )
  • Byes, ( 4 , 3 )
  • Cno
  • Dyes, ( 3 , 4 )
  • Eyes, ( 6 , 5 )

Q10:

The given figure shows the graphs of the functions 𝑓 ( π‘₯ ) = 3 π‘₯ βˆ’ 3 and 𝑔 ( π‘₯ ) = π‘₯ + 4 π‘₯ βˆ’ 5 2 . What are the points where 𝑓 ( π‘₯ ) = 𝑔 ( π‘₯ ) ?

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

Q11:

Find all values of π‘₯ where 𝑓 ( π‘₯ ) = 𝑑 ( π‘₯ ) , given 𝑓 ( π‘₯ ) = ( π‘₯ + 3 4 ) 2 and 𝑑 ( π‘₯ ) = π‘₯ + 3 4 .

  • A π‘₯ = βˆ’ 3 3 or π‘₯ = βˆ’ 3 4
  • B π‘₯ = βˆ’ 3 3 or π‘₯ = 3 5
  • C π‘₯ = βˆ’ 3 4 or π‘₯ = 3 3
  • D π‘₯ = βˆ’ 3 4 or π‘₯ = 3 5
  • E π‘₯ = βˆ’ 3 3 or π‘₯ = 3 3

Q12:

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

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

Q13:

Answer the following questions.

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

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

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

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

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

Q14:

Answer the following questions.

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

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

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

  • A 𝑦 = βˆ’ π‘₯ + 6
  • B 𝑦 = βˆ’ π‘₯ + 3
  • C 𝑦 = βˆ’ π‘₯ + 2
  • D 𝑦 = π‘₯ + 6
  • E 𝑦 = π‘₯ + 2

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

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

Q15:

The given diagram shows the graph of 𝑓 ( π‘₯ ) = ο€Ή π‘₯ βˆ’ 7 5 π‘₯  5 0 3 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 , βˆ’ 5 )
  • B ( βˆ’ 5 , 5 ) , ( 5 , 5 )
  • C ( 0 , 0 ) , ( 5 , 5 )
  • D ( βˆ’ 5 , βˆ’ 5 ) , ( 5 , 5 )
  • E ( βˆ’ 5 , 5 ) , ( 0 , 0 )

Note that from 𝑓 ( 2 ) = βˆ’ 7 1 2 5 , we know that line 𝑦 = 2 βˆ’ 7 1 2 5 βˆ’ π‘₯ meets 𝑦 = 𝑓 ( π‘₯ ) in the point ο€Ό 2 , βˆ’ 7 1 2 5  , so ( π‘₯ βˆ’ 2 ) is a factor of the cubic polynomial 𝑓 ( π‘₯ ) βˆ’ ο€Ό 2 βˆ’ 7 1 2 5 βˆ’ π‘₯  . Find the π‘₯ -coordinates of the other two intersections.

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

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

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

Q16:

Find all the possible values of π‘₯ satisfying 𝑓 ( π‘₯ ) = π‘₯ + 9 π‘₯ + 2 2 and 𝑓 ( π‘₯ ) = βˆ’ 1 6 given π‘₯ ∈ β„€ .

  • A βˆ’ 3 or βˆ’ 6
  • B3 or 6
  • C βˆ’ 6
  • D βˆ’ 3

Q17:

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

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