How to Sketch the Graphs of Linear Functions/Equations, ( f(x) = x )

This Article will show how to Sketch the graphs of Linear Functions by using only Two different values for ' x ',then finding the Coordinates of the Points through which the graph of the Equations/Functions are drawn, 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
• Graph Paper

Instructions

1. There is a Statement that says,... " The shortest distant between two Points is a straight line ".
   Based on that statement, we are going to use two Points to sketch the Graph of a Linear Function or a Linear Equation. Any Diagonal line and any Horizontal line can be classified as Linear Functions/Linear Equations, but any Vertical line can only be classified as a linear Equation. A vertical line cannot be classified as a Linear Function.

2. 

   The Equation of a Linear Function has the Form,...
   y = f(x) = Ax.
   If ( A ) equal to zero ( 0 ), the Shape of the Graph of the Linear Function is a Horizontal Line similar to, ' -- '.If ( A ) is greater than Zero ( 0 ), that is ( A ) is a Positive Number, The Shape of the Graph of the Linear Function is a Diagonal Line similar to, ' / '.
   If ( A ) is Less than Zero ( 0 ), that is ( A ) is a Negative Number, the Shape of the Graph is similar to, ' \ '. Please Click on the Image for a better view.

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

4. 

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

5. 

   Given the Equation y = f(x) = Ax + 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 Substitute x = ( 0 )
   into the Equation y = f(x) = Ax + B, that is y = f(0) = A( 0 ) + B = ( 0 ) + B= B, this is the First Point with Coordinates (0,B).Now Substituting x = ( 1 ) we get
   y = f(1) = A(1) + B = A + B, so the Second Point has Coordinates (1,A+B).. So the Graphs of the Diagonal Lines will pass through the First Point with Coordinates (0,B), and the Second Point
   that has Coordinates (1,A+B).. Please Click on the Image to get a better understanding.

6. 

   Given the Equation y = f(x) = A(x - B) 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. Please click on the Image for a better understanding.

7. 

   Given y = f(x) = A(x - B) + 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 #9. SO the First Point will have Coordinates (B,C), and the Second Point will have Coordinates (B+1,C). Please Click on the Image to get a better understanding.