How to Sketch the Graph of Cubic Functions, ( f(x)=x³ )

This Article will show how to Sketch the graph of a Cubic Function by using only three Points Derived from the Equation/Function itself, also it will show how the Graphs Vertically Translates ( moves up or down ), Horizontally Translates ( moves to the left or to the right ), and how the Graph simultaneously does Both Translations.

Things You'll Need

• Paper
• Pencil and
• Graph Paper

Instructions

1. 

   The Equation of a Cubic Function has the Form,... y = f(x) = Ax^3, where ( A ) must not be equal to zero ( 0 ).If ( A ) is greater than Zero ( 0 ), that is ( A ) is a Positive Number, The Shape of the Graph of the Cubic Function is similar to the reversed of the letter, ' $ '. If ( A ) is Less than Zero ( 0 ), that is ( A ) is a Negative Number, the Shape of the Graph is similar to that of an ' $ '. Please Click on the Image for a better view.

2. 

   To Sketch the Graph of The Equation,... y = f(x) = Ax^3, we choose Three Values for ' x ', x = ( -1 ), x = ( 0 ) and x = ( 1 ). We substitute each value of ' x ' into the Equation,... y = f(x) = Ax^3 and get the respective corresponding value for each ' y '.

3. Given y = f(x) = Ax^3, where ( A ) is a Real Number and ( A ) not equal to Zero ( 0 ), and substituting, x = ( -1 ) into the Equation we get y = f(-1) = A(-1)^3 = A(-1) = -A. So the First Point has Coordinates (-1,-A). Now Substituting, x = ( 0 ), we get y = f(0) = A(0)^3 = A(0)= 0. So the Second Point has Coordinates (0,0). And Substituting x = ( 1 ) we get y = f(1) = A(1)^3 = A(1) = A. So the Third Point has Coordinates (1,A). We now Sketch the Curve through these Three Points. Please Click on the Image for a better view.

4. 

   Given the Equation y = f(x) = Ax^3 + B, where B is any Real Number,
   the Graph of this Equation would Translate Vertically ( B ) units.
   If ( B ) is a Positive Number, the Graph will move up ( B ) units, and if ( B ) is a Negative Number, the Graph will move down ( B ) units. To Sketch The Graphs of this Equation, We follow the Instructions and use the same values of ' x ' of Step #3. Please Click on the Image to get a better view.

5. Given the Equation y = f(x) = A(x - B)^3 where A and B are any Real Numbers, and ( A ) not equal to Zero ( 0 ), the Graph of this Equation would Translate Horizontally ( B ) units.
   If ( B ) is a Positive Number, the Graph will move to the Right ( B ) units and if ( B ) is a Negative Number, the Graph will move to the Left ( B ) units. To Sketch The Graphs of this Equation, we First set the Expression,' x - B ', that is inside the parentheses, Equal to Zero, and solve for ' x '. That is,... x - B = 0, then x = B.

6. We will now use the following Three Values for ' x ', x = ( B - 1 ), x = ( B ) and x = ( B + 1 ). We substitute each value of ' x ' into the Equation,... y = f(x) = A(x - B)^3 and get the respective corresponding value for each ' y '.

7. 

   Given y = f(x) = A(x - B)^3, where A and B are Real Numbers, and ( A ) not equal to Zero ( o ). Substituting, x = ( B -1 ) into the Equation we get y = f(B-1) = A(B-1-B)^3 = A(-1)^3 = A(-1) = -A. So the First Point has Coordinates (B-1,-A). Now Substituting, x = ( B ), we get y = f(B) = A(B-B)^3 = A(0)^3 = A(0) = 0. So the Second Point has Coordinates (B,0). And Substituting x = ( B + 1 )we get y = f(B+1) = A(B+1-B)^3 = A(1)^3 = A(1) = A. So the Third Point has Coordinates(B+1,A). We now Sketch the Curve through these Three Points. Please Click on the Image for a better view.

8. 

   Given y = f(x) = A(x - B)^3 + C, where A, B, C are Real Numbers and ( A )not equal to Zero ( 0 ). If C is a Positive Number then the Graph in STEP #7 Will Translate Vertically ( C ) units.
   If ( C ) is a Positive Number, the Graph will move up ( C ) units, and if ( C ) is a Negative Number, the Graph will move down ( C ) units. To Sketch The Graphs of this Equation, We follow the Instructions and use the same values of ' x ' of Step #7. Please Click on the Image to get a better view.