mathematics
  Personalized »

KS3
 
 
 
Year 7
 
 
Year 8
 
 
Year 9

KS4
 
 
 
Year 10
 
 
Year 11

A-levels
 
 
 
Year 12 (AS)
 
 
Year 13 (A-levels)


physics
  Personalized »

 
 


KS4
 
 
 
KS4 Level Physics

A-levels
 
 
 
AS Level Physics
 
 
A-levels Physics


 
 
Private tutors
 
 
Find a tutor »


 


Unlock access to content
 
Select one option:



Free of charge
Unlock the chapter   "Graph of a Parabola"  



Unlock access to the solution of problems in this chapter.
Free of charge Price: 3 £

Continue »
Subscribe to OpenProf     If you do not get a better grade, you get your money back!
 


Unlock access to all problems solutions.
Available from 8.25 £ / month.
Continue »
mathematics UK Year 12 Mathematics (AS)
 
Solved problems for the chapter:
 

Graph of a Parabola

download pdf
View the theory »
Quadratic function
open
View the theory »
Vieta's formula
open
View the theory »
Quadratic equation
open
↓   Show theory [3]   ↓


Detailed solution steps are available at:
 
https://en.openprof.com/wb/chapter:graph_of_a_parabola/4692/


Example 1.

Given the quadratic function .


  • Write the given function in vertex form and draw its graph. Write down the range of values of the given function.

  • Shift the graph of the function f(x) such that it touches the abscissa axis and is symmetrical with respect to the ordinate axis. Write the equation of this new graph.

    
Example 2.

Given the quadratic function :


  • Calculate the zeros of f(x).


  • Find the coordinates of the vertex of the parabola f(x).


  • Find the point of intersection of the graph of the function f(x) with the y axis.


  • Draw the graph of the function f(x).


  • On what interval is the function f(x) negative?


    
3.

The figure shows the graph of the quadratic function f(x) passing through the points A, B and C.




  • Write down the function f(x).


  • Compute the coordinates of the vertex of the function f(x).


  • At exactly how many points does the graph of the function f(x) and the line with the equation intersect?


  • Find all values of the parameter for which the graph of the function f(x) and the line with the equation intersect at two distinct points.

    
4.

Given the :


  • Find the points of intersection of the coordinate axes and the graph of the given function.


  • Find the vertex of the function.


  • Draw the graph of the given function and write down for which values of x the function is negative.


  • Find the equation of the tangent to the curve at the point

   
5.

Klemen tried to graphically find the intersections of the line and the parabola, but he failed because he could not read the intersection point B exactly (see the graph below).


  • Find the equations of the line and the parabola in the graph.

  • Calculate the coordinates of the point B.


    
6.

The function f(x) is defined as .


  • Calculate the zeros, vertex point, and intersect on y-axis of the function f(x).

  • Draw the graph of the function f(x).

  • Calculate the intersection of the graph of the function f(x) and the line with the equation .

    
7.

  • Write the vertex form of a quadratic function that has it's vertex at point T (4, -5) and passes through the point (2, -1).


  • At what point does the graph of the function intersect the ordinate axis?

    
8.

Draw the graphs of the following functions in the same coordinate system:








    

9.

  • Write down the equation of the parabola that goes through points (1, - 6), (- 1, 0) and (2, - 24).


  • Does the point (-2, 24) lie on the parabola?


    
10.

Write the:

  • vertex form

  • general form

of a quadratic function whose vertex is at the point and its graph goes through the origin.

    
11.

A quadratic function f with the leading coefficient has and as its zeros. Write a relation for the function f and draw its graph. For what value c is the graph of the function symmetric with respect to the ordinate axis?

    
12.

In the set of parabolas , , find and write the equation of parabola whose vertex is on the x-axis.

    
13.

The figure below is a graph of a quadratic function.





From the graph, find:

  • The zeros,

  • coordinate of the vertex,

  • where the function is increasing,

  • where the function is positive.

    
14.

Given the function :


  • Write down the domain and range of the function.

  • Draw the graph of the function.

  • Find a rule for the inverse of the given function.

    
15.

Draw graphs of following functions on the same axes:








    
16.

Draw graphs of the following functions on the same axis:








    
17.

Plot the graphs of the following functions in the same coordinate system:








    
18.

The figure below shows a graph of the quadratic function f(x).




  • Write down the general and vertex form of the function f(x).


  • The function g(x) is defined by the rule g(x) = f(x- c). Determine the value of c such that the function g is symmetrical with respect to the ordinate axis, and write the vertex form of the function g(x).

    

19.

Given the quadratic function .


  • Find the value of m for which the function has its vertex at .

  • Write the general form of the function.

    
20.

Given the function :


  • Move the parabola with the equation so that the vertex of the new parabola will be at the point T(- 2, - 3).


  • Write the equation of the function g(x).

    
21.

A parabola with equation is given.


  • Reflect it over the abscissa, draw its graph and write down its equation.

  • Reflect it over the ordinate axis, draw its graph and write down its equation.

    
22.

Write the vertex and general form of the quadratic function f(x) whose graph is given below.


    
23.

Draw a parabola with the equation and mirror it over the coordinate axes. Draw the resulting curve in the coordinate system and write down its equation.

    
24.

  • Write the vertex form of a quadratic function that has its vertex at the point T (- 2, - 3) and its graph goes through the point (- 1, - 2).


  • Which of the points (1, 6), (-4, 1) and (3, 28) lie on the graph of the function f(x)?

    
25.

Given the quadratic function . Find the value of m for which the function has its vertex at the point (2, - 3).

    
26.

Given a parabola with a leading coefficient of 2 and a vertex at the point .


  • Write down the vertex and general form of the parabola.

  • If point A is the intersection of the parabola with the ordinate axis. Write down three forms of the equation of the line passing through point A and is parallel to the bisectors of the even quadrants.

    
27.

Given the quadratic function .


  • Write it in vertex form and draw its graph. Write down the range of the value of the given function.

  • Move the graph of the function f(x) in the direction of the abscissa axis so that it is symmetrical with respect to the ordinate axis. Write down the equation of the graph.

    
28.

The graph of the function g(x) is obtained by moving the graph of the function in parallel so that the vertex of the graph is at the point T(3, - 1).


  • Draw the graph of the functions f(x) and g(x).

  • Write the rule for the function g(x).

    

29.

Write the functions below in vertex form and draw their graphs.






    
30.

Given the graph of a quadratic function. Find the

  • roots of the function,

  • range of the function,

  • values of x for which the function is negative.


    
31.

Given the functions below.






  • Draw graphs of both functions in the coordinate system.


  • Calculate the points intersection of the graphs of the given functions.


  • Evaluate the value of .

    
32.

Graphs of the functions






are given below.

  • Calculate the intersections of the graphs f(x) and g(x).

  • Solve the inequality .



    
33.

Given the equation of the parabola




  • Draw the graph of the parabola.

  • What is the equation of the line parallel to the x-axis and passes through the vertex of the parabola?

  • Calculate the length of the line joining the points of intersections of the parabola and the line y = - 3.

    
34.

Write down all three forms of the quadratic function whose graph is in the figure:


    
35.

Write down all three forms of the quadratic function whose graph is in the figure:


    
36.

The graph of the function f is given.




  • Write down the expression for the function f.

  • Draw graphs of the functions and on the same coordinate system.

  • Move the graph of the function f in the direction of the abscissa axis so that the function g: , whose graph you got, is even. Write down the number c and the expression for the function g.

    
37.

Draw the graph of the functions and .

    
38.

From the set of parabolas below write and draw the one touching the abscissa axis:


    

39. Make the necessary corrections

The quadratic function is given, where m is any real number. For write the rule of the function f and draw its graph .

    
40. The figures below show the graphs of the functions f, g and h. Write down their regulations. <= Change the word "regulation" there. You can replace it with "expressions"





    
41.

The function is given:


  • Write down the zeros, the intersection with the ordinate axis and the coordinates of the vertex of the function f.


  • Draw a graph of the function f exactly.


  • Write the equation of the symmetric axis of the parabola.


  • For which number n will the line be tangent to the graph of the function f?


    
42.

For which value of the parameter n will be the line of the tangent to the graph of the function ? Draw a graph of the function f and the required tangent .

    
43.

The quadratic functions and are given.


  • Draw their graphs in the same coordinate system.

  • Write down all three forms of the equation of the line passing through the intersections of the graphs of the two functions.

    
44.

The function is given. In the same coordinate system, draw a graph of the function f and its tangents intersecting the ordinate axis at .