Worksheet: Finding the Intersection Points of Two Functions

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

Q1:

Find the set of points of intersection of the graphs of 𝑦 = 3 𝑥 and 𝑥 + 𝑦 = 4 0 2 2 .

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

Q2:

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

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

𝑥 2 1 0 1 2
𝑦
  • A 6 , 0 , 3 , 7 , 8
  • B 4 , 2 , 4 , 6 , 6
  • C 4 , 2 , 3 , 7 , 8
  • D 6 , 0 , 4 , 6 , 6
  • E 4 , 2 , 3 , 6 , 6

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

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

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

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

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

  • Ayes, ( 5 , 6 )
  • Byes, ( 4 , 3 )
  • Cyes, ( 3 , 4 )
  • Dno
  • Eyes, ( 6 , 5 )

Q3:

Answer the following questions.

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

  • A 𝑦 = 2 𝑥 4
  • B 𝑦 = 2 𝑥 4
  • C 𝑦 = 𝑥 + 2
  • D 𝑦 = 2 𝑥 + 4
  • E 𝑦 = 𝑥 + 1

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

  • A 𝑦 = 2 𝑥 2
  • B 𝑦 = 2 𝑥 4
  • C 𝑦 = 2 𝑥 4
  • D 𝑦 = 2 𝑥 + 2
  • E 𝑦 = 2 𝑥 + 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).
  • CNo, they do not intersect.
  • DYes, they intersect at ( 0 , 2 ) .
  • EYes, they intersect at ( 1 , 0 ) .

Q4:

Answer the following questions.

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

  • A 𝑦 = 𝑥 + 1
  • B 𝑦 = 2 𝑥 + 3
  • C 𝑦 = 𝑥 + 3
  • D 𝑦 = 2 𝑥 + 3
  • E 𝑦 = 4 𝑥 + 7

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

  • A 𝑦 = 3 𝑥 2
  • B 𝑦 = 3 𝑥 4
  • C 𝑦 = 3 𝑥 + 2
  • D 𝑦 = 3 𝑥 + 2
  • E 𝑦 = 3 𝑥 + 4

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

  • ANo, they do not intersect.
  • BYes, they intersect at ( 2 , 4 ) .
  • CYes, they intersect at ( 1 , 1 ) .
  • DYes, they intersect at ( 0 , 3 ) .
  • EYes, they intersect at ( 2 , 1 ) .

Q5:

Answer the following questions.

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

  • A 𝑦 = 2 𝑥 4
  • B 𝑦 = 2 𝑥 8
  • C 𝑦 = 𝑥 + 2
  • D 𝑦 = 𝑥 2
  • E 𝑦 = 2 𝑥 2

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

  • A 𝑦 = 𝑥 + 6
  • B 𝑦 = 𝑥 + 6
  • C 𝑦 = 𝑥 + 2
  • D 𝑦 = 𝑥 + 2
  • E 𝑦 = 𝑥 + 3

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

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

Q6:

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

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

Q7:

The given figure shows the graphs of the functions 𝑓 ( 𝑥 ) = 6 and 𝑔 ( 𝑥 ) = 3 𝑥 3 𝑥 2 . What are the points where 𝑓 ( 𝑥 ) = 𝑔 ( 𝑥 ) ?

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

Q8:

The given figure shows the graphs of the functions 𝑓 ( 𝑥 ) = 3 𝑥 3 and 𝑔 ( 𝑥 ) = 𝑥 + 4 𝑥 5 2 . What are the points where 𝑓 ( 𝑥 ) = 𝑔 ( 𝑥 ) ?

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

Q9:

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

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

Q10:

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

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

Q11:

The given diagram shows the graph of 𝑓 ( 𝑥 ) = 𝑥 7 5 𝑥 5 0 3 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 ) , ( 0 , 0 )
  • B ( 0 , 0 ) , ( 5 , 5 )
  • C ( 5 , 5 ) , ( 5 , 5 )
  • D ( 5 , 5 ) , ( 5 , 5 )
  • E ( 5 , 5 ) , ( 5 , 5 )

Note that from 𝑓 ( 2 ) = 7 1 2 5 , we know that line 𝑦 = 2 7 1 2 5 𝑥 meets 𝑦 = 𝑓 ( 𝑥 ) in the point 2 , 7 1 2 5 , so ( 𝑥 2 ) is a factor of the cubic polynomial 𝑓 ( 𝑥 ) 2 7 1 2 5 𝑥 . Find the 𝑥 -coordinates of the other two intersections.

  • A 2 2 1 , 2 2 1
  • B 2 2 1 , 2 2 + 1
  • C 2 2 + 1 , 2 2 + 1
  • D 2 2 + 1 , 2 2 1
  • E 2 2 , 2 2

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 .

  • A 4 𝑑 + 2 𝑐 𝑑 + 𝑐 2 5 = 0 2 2
  • B 4 𝑑 + 2 𝑐 𝑑 + 𝑐 2 5 = 0 2 2
  • C 𝑑 + 𝑐 𝑑 + 𝑐 2 5 = 0 2 2
  • D 𝑑 + 𝑐 𝑑 + 𝑐 = 0 2 2
  • E 𝑑 𝑐 𝑑 + 𝑐 2 5 = 0 2 2

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 .

  • A 𝑐 = 1 0 3 3 , 𝑑 = 5 3 3
  • B 𝑐 = 3 , 𝑑 = 3 2
  • C 𝑐 = 1 0 3 3 , 𝑑 = 5 3 3
  • D 𝑐 = 1 0 3 , 𝑑 = 5 3
  • E 𝑐 = 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 𝑥 = 𝑐 to 𝑥 = 𝑑 is 1 ?

  • A 𝑐 < 3 3 or 𝑐 > 3 3
  • B 𝑐 < 1 0 3 3 or 𝑐 > 1 0 3 3
  • C 𝑐 > 3 3 or 𝑐 < 3 3
  • D 𝑐 > 1 0 3 3 or 𝑐 < 1 0 3 3
  • E 𝑐 > 1 0 3 or 𝑐 < 1 0 3

Q12:

Find all values of 𝑥 where 𝑓 ( 𝑥 ) = 𝑡 ( 𝑥 ) , given 𝑓 ( 𝑥 ) = ( 𝑥 + 3 4 ) 2 and 𝑡 ( 𝑥 ) = 𝑥 + 3 4 .

  • A 𝑥 = 3 3 or 𝑥 = 3 3
  • B 𝑥 = 3 4 or 𝑥 = 3 3
  • C 𝑥 = 3 4 or 𝑥 = 3 5
  • D 𝑥 = 3 3 or 𝑥 = 3 4
  • E 𝑥 = 3 3 or 𝑥 = 3 5

Q13:

Find all the possible values of 𝑥 satisfying 𝑓 ( 𝑥 ) = 𝑥 + 9 𝑥 + 2 2 and 𝑓 ( 𝑥 ) = 1 6 given 𝑥 .

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

Q14:

The graphs of 𝑓 1 and 𝑓 2 intersect at the point ( 𝑘 , 2 ) , where 𝑓 ( 𝑥 ) = 2 1 𝑥 and 𝑓 ( 𝑥 ) = 3 𝑥 2 . Find the set of possible values of 𝑘 .

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

Q15:

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

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

Q16:

Find the 𝑥 -intercepts of the function 𝑓 ( 𝑥 ) = 3 𝑥 1 5 𝑥 4 2 .

  • A 2 and 7
  • B2 and 7
  • C2 and 7
  • D 2 and 7

Q17:

Find the set of points of intersection of the graphs of 𝑥 = 𝑦 and 𝑥 + 𝑦 = 3 2 2 2 .

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

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