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Linear relationships are used in everyday life.These relationships can be expressed in many different ways. Linear graphs are one way of expressing these relationships, when graphed they give a straight line. Linear graphs can be sketched or plotted. Plotting a Straight Line Graph Sketching a Straight Line Graph Key Facts about Graphs Gradient between two points         Plotting a straight line graph: Use  graph paper or grid paper for accuracy. Use X and Y as variables. Give values to X and work out the values of Y, set these values in a table. Place the axes in the most suitable position, to give the best presentation of the graph. Choose a suitable scale. Plot the points from your table of values. Join the points to give a straight line.     Sketching a straight line graph: Graph paper or grid paper are not necessary. Only two points are necessary to sketch a straight line graph. Join the two points to give a straight line.   There are different ways of writing   a linear equation or an equation of a straight line. y = mx + c , where m is the gradient and c is the y-intercept. ax + by + c = 0, where a, b, c are constants. y-y1 = m (x - x1), where m is the gradient and x1 and y1 are co-ordinates of a given point. If you are not sure about the terms, go to Graphs- basic terms.       Key facts about the gradients and the shape of the straight line graphs:       .

 

 

 

 

 

 

 

Examples

 

 

Example 1: Draw the graph of the linear equation  y = 2x +3

Example 2: Sketch the linear equation y = 2x +3 , using the x- and y- intercepts.

Example 3: Rearrange the linear equation,  3y + 4x = 12   to  y= mx + c   form and find the y- intercept .

 

 

 

 

 

 

 

Example 1. Draw the graph of the linear equation  y = 2x +3 First make a table. Let's choose x values between -3 and +3 row1 x -3 -2 -1 0 1 2 3 (1) 2×row1 2x -6 -4 -2 0 2 4 6 (2) add 3 +3 +3 +3 +3 +3 +3 +3 +3 (1)+(2),  y = 2x+3 -3 -1   1   3   5   7   9 Plot these points on a number plane as shown below. Return to Examples   Return to Examples     Example 2. Sketch the linear equation y = 2x +3 , using the x- and y- intercepts.    To find the x-intercept, let y =0    When y =0,   2x + 3 = 0                                2x = -3                                 x = -3 ÷ 2 = -1.5    Now we can write the co-ordinates of the x-intercept as (-1.5,0)   To find the y- intercept, let x =0   When x = 0, y = 2×0 +3                       y = 3 Now we can write the co-ordinates of the y-intercept as (0,3) Return to Examples We can join  (-1.5,0) and (0,3) to sketch the graph.   Return to Examples         Example 3. Rearrange the linear equation,  3y + 4x = 12   to  y= mx + c   form and find the y- intercept . 3y+4x =12 3y    = -4x + 12   y  = (-4x +12) ÷ 3   y  = (-4x ÷ 3)  +(12 ÷ 3) Return to Examples     Leave the y term on the left hand side(L.H.S), and take all the other terms to the   right hand side(R.H.S.), now you have 3y = -4x +12 Then divide the whole equation by 3 to give y by itself. Now you have rearrange the equation to y = mx + c form. So the gradient is m = -1.33 y-intercept , which is c =4

 

 

 

 

 

 

Have a Go Problem 1 Find the gradient between A(2,5) and B(4,9) and then find the equation of the straight line.     Problem 2 Find the X- and Y intercepts for y =5x -3 and hence sketch the graph.

 

 

 

 

 

 

Practice Questions Question 1 Find the gradient between A(1,-4) and B(-1,4)   Answer m = - 4 Question 2 State the gradient and y-intercept of 2y = 6 x - 8   answer m=3, y intercept (0, -4)   Question 3 Find the gradient between A(4,2) and B(-3,2) and hence find the equation of line AB.   answer m=0, y=2   Question 4 Sketch  3y = -6 x +15 click here to check your answer

 

 

 

 

 

 

Solution 1 Find the gradient between A(2,5) and B(4,9) and then find the equation of the straight line. Let A(2,5) =(x1,y1) and B(4,9) =(x2,y2) m = 2 and  (x1,y1) =(2,5) Equation of the straight line is: back to Have a Go                  y -  y1 = m(x - x1)                           y - 5   = 2 (x - 2)                   y - 5  = 2x -4                      y     = 2x -4 +5  (add 5 to both sides, on L.H.S. -5 +5=0)                      y    =  2x + 1

Solution 2 Find the X- and Y intercepts for y =5x -3 and hence sketch the graph.   To find the x-intercept, let y =0    When y =0,   5x -3 = 0                                5x = 3                                 x = 3 ÷ 5 = 0.6    Now we can write the co-ordinates of the x-intercept as (0.6,0)   To find the y- intercept, let x =0   When x = 0, y = 5×0 -3                       y = -3   Now we can write the co-ordinates of the y-intercept as (0,-3) We can join the points (0.6,0) and (0,-3) to sketch the graph.   back to Have a Go

 

                                       

 

Answer to Practice Question 4

 

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