OCR Advanced FSMQ - Additional Maths

Suppose you are asked to draw the line with equation y = 2 x – 3. The quickest way to do this is to find where the line cuts the x - and y -axes. To find where the line cuts the y -axis, we substitute x = 0 (which is the equation of the y -axis) into the equation of the line. Hence, y = 2(0) – 3, giving y = –3. To find where the line cuts the x -axis, we substitute y = 0 (which is the equation of the x-axis) into the equation of the line. Hence, 0 = 2 x – 3, giving 2 x = 3, so x = 3 2 . These two points (0, –3) and ( 3 2 , 0) are plotted on a set of axes and the line is drawn passing through each of them. Remember to label the axes and add the equation to the line. y x O –1 –2 –3 1 2 3 2 y = 2 x – 3 Gradient = y 2 − y 1 x 2 − x 1 = 10 − 0 6 − (− 2) = 10 8 = 5 4 6.5 The equation of a straight line graph Straight line graphs are also called linear graphs and have an equation of the form y = mx + c Notice there is one y on the left-hand side of the equation. m is the gradient (i.e. the steepness of the line) and c is the intercept on the y -axis. It is important that in this equation there is only an x term, so there are no terms containing x 2 , x 3 , √ x , 1 x , etc. The following equation is an equation of a straight line: y = 2 x − 3 Comparing this equation with y = mx + c you can see that the gradient, m is 2 and intercept on the y -axis, c is −3. Examples 4 Which of each of the following equations are equations of straight lines? (a) y = 1.4 x + 7 (b) y = 4 x 2 + 1 (c) y = 2 x (d) y = −4 x (e) y = √ x (f) 2 y = 8 x + 6 (g) 5 x + 2 y = 3 (h) y = 4 3  x − 2 Answers 4 (a) Yes, y = 1.4 x + 7 is in the form y = mx + c (b) No, y = 4 x 2 + 1 is not in the form y = mx + c as it contains a term in x 2 . (c) No, y = 2 x is not in the form y = x + c (d) Yes, y = −4 x is in the form y = mx + c with c = 0. (e) No, y = √ x is not in the form y = mx + c (f) Yes, 2 y = 8 x + 6 can be divided by 2 to give y = 4 x + 3 which is in the form y = mx + c y 0 A (–2, 0) B (6, 10) x Midpoint M = ( –2 + 62  , 0 + 10 2 ) = (2, 5) M 6 Coordinate geometry of straight lines 123