LINEAR FUNCTIONS LINEAR FUNCTIONS A linear function is of the type y = mx + c. Examples of linear functions are: y = 4x, y = 5 – 3x, 𝑥 𝑥 𝑦 = 2, 4 𝑦 +5=2 For pairs of values of x and y as shown below, we can say it is linear if the first differences are same. x y 2 7 First difference 3 10 4 13 3 3 5 16 3 What is the rule? The rule is y = 3x + 1. State which of the following is linear functions. Function Linear or Not y = 2x – 1 𝑦= YES 3 𝑥 4 y = 7 + 4x y2 = 2x 5 y=𝑥 𝑦 = 𝑥2 + 4 𝑥 = 𝑦−4 2 Page 1 6 19 3 LINEAR FUNCTIONS EXERCISE 3.1 Investigate which of the following are linear functions. If yes, find the rule without a calculator. (a) x y 1 3 2 5 3 7 4 9 5 11 6 13 x y -1 7 0 10 1 13 2 16 3 19 4 22 x y 3 10 5 15 7 20 9 31 11 40 0 10 1 8 2 6 3 4 (b) (c) (d) x y Page 2 LINEAR FUNCTIONS Tabulate Use your calculator to complete a table of values. 1. Complete the following tables using your calculator. x y = 2x -2 -4 -1 0 0 1 2 5 10 x y = 3x -3 -9 -2 0 0 1 4 12 10 x y = 7 - 2x 0 7 1 2 3 1 4 5 x y = 4x + 1 2 3 13 5 8 33 10 20 x y = 8 - 3x 0 1 2 2 3 4 5 -7 2. Complete the following tables using your CAS calculator. x y = 5x - 2 2 8 3 5 8 38 10 20 x y = 3x - 5 -2 -11 -1 0 1 3 5 x y=x+5 -4 1 -3 -2 3 0 5 8 Page 3 LINEAR FUNCTIONS FINDING GRADIENT AND Y-INTERCEPT Express each of the following in the form y = mx + c and hence state the gradient (m) and the y-intercept (c). Remember to obtain y by itself on the left-hand side, then pick up the value of m and c. Equation Gradient (m) y = 2x + 5 y = 3x + 7 y=x-3 y = 0.5x + 1 y = 5x + 4 y=x+6 y = 4x – 1 y = -x + 9 y = 7x – 5 2y = 4x + 7 Divide by 2 Y = 2x + 3.5 3y = 7x – 6 Divide by 3 7 y = x2 3 2y = 8x + 5 2 3 1 0.5 y-intercept (c or b) 5 7 -3 1 2 3.5 7 3 -2 -1 5 2y = 6x – 1 3y = 4x + 3 5y = 2x + 3 y+x=5 y =-x + 5 y + 3x = 7 2y – 5x = 3 Page 4 LINEAR FUNCTIONS FINDING GRADIENT To find the gradient (m) of a line joining two points A( x1 , y1 ) and B( x2 , y2 ) , use the formula m= 𝒚𝟏 −𝒚𝟐 𝒙𝟏 −𝒙𝟐 Find the gradient (m) of the line joining the points. Points (2,3) and (4, 7) (3, 6) and (1, -4) Gradient (m) 3−7 −4 m = 2−4 = −2= 2 m= (1,2) and (5, 6) (1, 3) and (3, 9) (-1,6) and (3, 10) (-2, 5) and (3, 10) (0,2) and (3, 5) (2, -3) and (1, 0) (1,2) and (8, 10) (5, 4) and (3, -2) Page 5 6 (4) 10 = =5 3 1 2 LINEAR FUNCTIONS EQUATION OF LINE : GIVEN GRADIENT AND Y-INTERCEPT The equation of a line is given by y = mx + c, where m is the gradient and c is the y– intercept. To find the equation of a line we need 1. the gradient 2. the y-intercept Example 1 Find the equation of the line having gradient 5 and crossing the y-axis at 3. Solution: y = 5x + 3 Example 2 Find the equation of the line having slope -6 and y-intercept 4. Solution: y = -6x + 4 EXERCISE 1. Find the equation of the line having gradient 7 and crossing the y-axis at 2. 2. Find the equation of the line having gradient 3 and crossing the y-axis at -2. 3. Find the equation of the line having slope -1 and y-intercept 6. 4. Find the equation of the line having gradient 9 and crossing the y-axis at 7. 5. Find the equation of the line having slope -2 and y-intercept 10. 6. Find the equation of the line having slope -5 and y-intercept 1. 7. Find the equation of the line having gradient 1 and crossing the y-axis at (0,5). 8. Find the equation of the line having gradient 4 and crossing the y-axis at (0,-3). Page 6 LINEAR FUNCTIONS Equation of a line : GRADIENT GIVEN AND A POINT Example Find the equation of the line having gradient 2 and passing through A(3,5). Solution: Let 𝑦 − 𝑦1 = 𝑚(𝑥 − 𝑥1 ) be the equation of the line m = 2 x1 = 3 and y1 = 5 𝑦 − 5 = 2(𝑥 − 3) 𝑦 − 5 = 2𝑥 − 6 𝑦 = 2𝑥 − 6 + 5 𝑦 = 2𝑥 − 1 EXERCISE 1. Find the equation of the line having gradient 3 and passing through (4,5). 2. Find the equation of the line having gradient 5 and passing through (-2,4). 3. Find the equation of the line having slope 6 and passing through (-1,3). 4. Find the equation of the line having gradient -4 and passing through (2,1). 5. Find the equation of the line having gradient -5 and passing through (-2,3). Page 7 LINEAR FUNCTIONS HOW TO SHOW A POINT LIES ON A LINE? To show a point lies on a given line, replace the x and y value in the equation of the line we have to obtain the constant value Examples Show that the point A(2,5) lies on the line with equation 4x + y = 13. Solution Replace x = 2 and y = 5 in the equation. We have to obtain the right hand side number. 4(2) + (5) = 13 (shown) Does the point B(3,-2) lie on the line with equation 2x + y = 10? Solution Replace x = 3 and y = -2 in the equation. 2(3) + (-2) = 4 10 Therefore B does not lie on the line. The point A(p,3) lies on the line 4x – 2y = 2, find the value of p. Solution Replace x by p and y by 3 in the given equation and solve for p 4(p) – 2(3) = 2 4p – 6 = 2 4p = 8 p=2 EXERCISE 1. Show that the point (4,3) lies on the line 2. Show that the point (5,-1) lies on the with equation 3x + 2y = 18. line with equation x + 2y = 3. 3. Does the point (2,5) lie on the line with equation 3x - y = 1? 4. The point A(t,4) lies on the line 4x – 2y = 12, find the value of t. 5. The point A(4,m) lies on the line 5x + 2y = 6, find the value of m. 6. The point A(q,4) lies on the line 3x – 5y = -2, find the value of q. Page 8 LINEAR FUNCTIONS DISTANCE BETWEEN TWO POINTS To find the distance (d) between two points A( x1 , y1 ) and B( x2 , y2 ) , use the formula d = √(𝑥1 − 𝑥2 )2 + (𝑦1 − 𝑦2 )2 Example Find the distance between the points A(5,7) and B(2,3). Solution D= ( x 2 x1 ) 2 ( y 2 y1 ) 2 = √(5 − 2)2 + (7 − 3)2 = √32 + 42 = 5 units EXERCISE 1. Find the distance between the points A(1,2) and B(5,5). 2. Find the distance between the points P(-1,2) and Q(5,10). 3. Find the distance between the points A(-2,3) and B(10,8). 4. Find the distance between the points L(0,-3) and M(24,4). 5. Find the distance between the points A(1,4) and B(3,6). 6. Find the distance between the points P(-2,-3) and Q(4,5). Page 9 LINEAR FUNCTIONS PARALLEL LINES When two lines are parallel, they have the same gradient. For example, y = 2x, y = 2x - 3 and y = 2x + 5 are parallel. We can also say that these lines belong to the family of lines having gradient 2. Similarly, y = 3x and 2y = 6x -1 are parallel because both lines have a slope of 3. CLASS ACTIVITY 1 Two lines from each of these groups are parallel. Name them. Group 1 A : y 3x 2 Gradient Group 2 P : y 5x 2 Gradient Group 3 L : y 4x 2 C : 3y 9x 5 Q : 2 y 12 x 1 R : 3 y 12 x 5 M : 2 y 8x 1 N : 3 y 15 x 1 D : y 2x 4 S : y 5x 7 P : y 4x 3 B : 2 y 8x 1 Gradient Group 1 : Group 2 : Group 3 : CLASS ACTIVITY 2 Which of the following are parallel? There may be more than two lines parallel. Group 4 A : y 2x 3 B : 2 y 4x 1 C : 3y 6x 5 D : y 2x 9 Gradient Group 5 P: y x2 Q : 2 y 4x 1 R : 3 y 3x 4 S:yx3 Gradient Group 4 : Group 5 : Group 6 : Page 10 Group 6 L : y 7x 2 M : 2 y 14 x 1 N : 3 y 21x 1 P : y 7x 0 Gradient LINEAR FUNCTIONS Equation of a line passing through a point and parallel to a given line Example 1 Find the equation of the line passing through A(3,5) and parallel to y = 2x + 7. Solution: The gradient of the given line is 2. Therefore, the gradient of the required line will be 2 as well. Let 𝑦 − 𝑦1 = 𝑚(𝑥 − 𝑥1 ) be the equation of the line 𝑦 − 5 = 2(𝑥 − 3) 𝑦 − 5 = 2𝑥 − 6 𝑦 = 2𝑥 − 6 + 5 ∴ 𝑦 = 2𝑥 − 1 EXERCISE 1. Find the equation of the line passing through (2,3) and parallel to 𝑦 = 4𝑥 + 11. 2. Find the equation of the line passing through (-4,5) and parallel to 𝑦 = 3𝑥 − 7. 3. Find the equation of the line passing through (1,-2) and parallel to 𝑦 = 5𝑥 + 8. 4. Find the equation of the line passing through (0,6) and parallel to 2𝑦 = 6𝑥 − 7. 5. Find the equation of the line passing through (1,5) and parallel to 3𝑦 = 6𝑥 − 5. 6. Find the equation of the line passing through (-2,7) and parallel to 𝑦 − 5𝑥 = 4. Page 11 LINEAR FUNCTIONS Perpendicular lines When two lines are perpendicular, the product of their gradients is -1. 1 1 For example, y = 2x and y = x + 5 are perpendicular because 2 × − 2 = −1 2 3 4 3 4 Similarly, y = x + 1 and y = x + 5 are perpendicular because 4 × − 3 = −1 3 4 CLASS ACTIVITY Copy and complete the following table. Equation of line Gradient of parallel line y = 4x m=4 3 x6 5 y = 3x m= y= 3 5 y = 5x – 3 y= 2 x6 5 y = -6x + 5 y = 7x 2 y= 3 x 1 8 y = x3 5 y = x2 3 Page 12 Gradient of perpendicular line (flip and change sign) 1 m=4 5 m=3 LINEAR FUNCTIONS How to show two lines are perpendicular? To show two lines are perpendicular, simply multiply the gradients of both lines and the result must be -1. Examples Show that the line L1 : 𝑦 = 4𝑥 + 5 and the −1 line L2 : 𝑦 = 4 𝑥 + 3 are perpendicular. Show that the lines 2𝑦 = 6𝑥 + 1 and 3𝑦 + 𝑥 = 7 are perpendicular. Solution Solution Gradient of L1 = 4 Gradient of 2𝑦 = 6𝑥 + 1 is 3 1 Gradient of 3𝑦 + 𝑥 = 7 is − 3 1 Gradient of L2 = − 4 Product of gradients = 4 × −1 4 = −1 (𝑠ℎ𝑜𝑤𝑛) Product of gradients = 3 × −1 3 = −1 (𝑠ℎ𝑜𝑤𝑛) Are the lines 𝑦 = 5𝑥 − 3 and 2𝑦 = 𝑥 + 4 are perpendicular? Solution Gradient of 𝑦 = 5𝑥 − 3 is 5 Gradient of 2𝑦 = 𝑥 + 4 is 1 2 1 Product of gradients = 5 × 2 ≠ −1 Hence the two lines are NOT perpendicular. EXERCISE 1. Show that the line L1 : 𝑦 = 2𝑥 + 5 and −1 the line L2 : 𝑦 = 2 𝑥 + 3 are perpendicular. 2. Show that the lines 2𝑦 = 8𝑥 − 1 and 4𝑦 + 𝑥 = 5 are perpendicular. 3. Are the lines 𝑦 = 2𝑥 − 3 and 2𝑦 = 𝑥 + 5 are perpendicular? 4. Are the lines 2𝑦 = 5𝑥 + 1 and 5𝑦 = −2𝑥 are perpendicular? Page 13 LINEAR FUNCTIONS Equation of a line passing through a point and perpendicular to a given line. Example 1 Find the equation of the line passing through A(6,5) and perpendicular to y = 2x + 7. Solution: 2 The gradient of the given line is 2 or 1. Therefore the gradient of the required line will be – 1 . (flip and change sign) 2 Let 𝑦 − 𝑦1 = 𝑚(𝑥 − 𝑥1 ) be the equation of the line 1 𝑚 = − 2 , 𝑥1 = 6 𝑎𝑛𝑑 𝑦1 = 5 1 𝑦 − 5 = − (𝑥 − 6) expand the bracket 2 1 𝑦 − 5 = −2𝑥 + 3 add 5 on both sides 1 ∴ 𝑦 = −2𝑥 + 8 EXERCISE 1. Find the equation of the line passing through A(4,3) and perpendicular to 𝑦 = 4𝑥 + 11. 2. Find the equation of the line passing through A(9,-5) and perpendicular to 3 𝑦 = − 4 𝑥 + 8. 3. Find the equation of the line passing through A(10,7) and perpendicular to 5 𝑦 = 3 𝑥 + 4. 4. Find the equation of the line passing through A(6,-5) and perpendicular to 2𝑦 = 6𝑥 + 1. Page 14 LINEAR FUNCTIONS Equation of a line passing through two points To find the equation of a line passing through two points we find y y1 1. the gradient by using 2 x 2 x1 2. the y-intercept, by choosing any one of the two given points. Example 1 Find the equation of the line passing through A(3,1) and B(5,9) Solution: y y1 Since the gradient is not given, we use the formula m = 2 to find m. x 2 x1 9−1 8 𝑚 = 5−3 = 2 = 4 From the two points A and B, we can choose any point as the final answer would be same. Let us choose A (3,1) to find the y-intercept c. Let 𝑦 − 𝑦1 = 𝑚(𝑥 − 𝑥1 ) be the equation of the line m = 4 x1 = 3 and y1 = 1 𝑦 − 1 = 4(𝑥 − 3) 𝑦 − 1 = 4𝑥 − 12 𝑦 = 4𝑥 − 12 + 1 𝑦 = 4𝑥 − 11 If we choose B(5,9) instead, then m = 4 x1 = 5 and y1 = 9 𝑦 − 9 = 4(𝑥 − 5) 𝑦 − 9 = 4𝑥 − 20 𝑦 = 4𝑥 − 20 + 9 𝑦 = 4𝑥 − 11 same as before EXERCISE 1. Find the equation of the line passing through A(2,5) and B(6,9). 2. Find the equation of the line passing through A(-2,3) and B(1,9). 3. Find the equation of the line passing through P(0,2) and B(2,8). 4. Find the equation of the line passing through A(-3,1) and B(1,5). Page 15 LINEAR FUNCTIONS How to sketch a line? To sketch a line, we can either use a table of values or the intercept-method. Example 1 Sketch the line y = 2x + 3. Solution We make a table as under. Note: two pairs of values is enough. x y = 2x + 3 0 3 1 5 Plot these two co-ordinates on the Cartesian axes to sketch the line. EXERCISE a Sketch the line y = x + 3. b Sketch the line y = 3x – 1. y -4 y 10 10 8 8 6 6 4 4 2 2 -2 2 4 6 8 10 x -4 -2 2 -2 -2 -4 -4 Page 16 4 6 8 10 x LINEAR FUNCTIONS c Sketch the line y = 4 – x. d Sketch the line y = x – 2. y y 10 10 8 8 6 6 4 4 2 -4 2 -2 2 4 6 8 10 x -4 -2 -2 2 4 6 8 x 10 -2 -4 -4 e Sketch the line y = 5 – 2x. f Sketch the line y = 3x + 2. y -4 g y 10 10 8 8 6 6 4 4 2 2 -2 2 4 6 8 10 x -4 -2 2 -2 -2 -4 -4 Sketch the line y = 8 – 2x. h 8 10 x y 10 10 8 8 6 6 4 4 2 2 -2 6 Sketch the line y = 5x + 2. y -4 4 2 4 6 8 10 x -2 -4 -2 2 -2 -4 -4 Page 17 4 6 8 10 x LINEAR FUNCTIONS Horizontal and Vertical lines Horizontal lines are of the type y = a, where a is any number. Vertical lines are of the type x = b, where b is any number. Example x = -4 y x=3 10 y=5 5 -10 -5 5 x 10 -5 y = -8 -10 EXERCISE 1. On the same set of axes sketch the line y = 4, y = -1, and x = 2. y 10 8 6 4 2 -4 -2 2 -2 -4 Page 18 4 6 8 10 x LINEAR FUNCTIONS 2. On the same set of axes sketch the line x = 3, x = -1, and y = 3. y 10 8 6 4 2 -4 -2 2 4 6 8 x 10 -2 -4 3. On the same set of axes sketch the line x = 5, x = -4, and y = 7. y 10 8 6 4 2 -4 -2 2 4 6 8 10 x -2 -4 4. On the same set of axes sketch the line x = 9, x = -2, and y = 6. y 10 8 6 4 2 -4 -2 2 -2 -4 Page 19 4 6 8 10 x LINEAR FUNCTIONS How to find the equation of a line given the sketch? Example Find the equation of the following line. (a) (b) Solution This line has a y-intercept of 2. The gradient can be obtained by using any two points on the line. (0,2) and (-2,0) 02 2 1 m= 20 2 So the equation is y = x + 2 Alternately, we use gradient is rise over run to find the equation. Page 20 Solution This line has a y-intercept of 6. The gradient can be obtained by using any two points on the line. (0,6) and (2,0) 06 6 3 m= 20 2 So the equation is y = -3x + 6 LINEAR FUNCTIONS 10 2 4 6 8 – 4 2 2– 4 4 6 8 10 2 EXERCISE 10 2 4 6 8 – 4 2 2– 4 4 6 8 10 2 Find the equations of the following lines y y 10 10 8 8 6 6 4 4 2 – 4 2 – 2 2 4 6 8 10 x – 4 – 2 10 2 4 6 8 – 4 2 2– 4 4 6 8 10 2 10 2 4 6 8 – 4 2 2– 4 4 6 8 10 2 – 2 2 4 6 10 x 8 – 2 – 4 – 4 y y – 4 10 10 8 8 6 6 4 4 2 2 – 2 10 2 4 6 8 – 4 2 2– 4 4 6 8 10 2 2 4 6 8 10 – 2 x 10 2 4 6 8 – 10 2 4 6 8 2– 4 4 2 – 4 – 2 2 4 6 10 x 8 – 2 – 4 – 4 y y 10 10 8 8 6 6 4 4 2 2 – 4 – 2 2 – 2 – 4 – 2 2 4 6 8 10 x – 4 – 2 – 6 – 8 – 4 – 10 Page 21 4 x LINEAR FUNCTIONS FUNCTIONS In this section, we are going to have a look at functions such as f(x), read as f of x. In the first section, the reader would be able to understand and try to find the image of a number. Example Given that f(x) = 3x + 5, find (a) f(4) (b) f(-6) Solution (a) f(4) = 3(4) + 5 [replace the x by 4, and evaluate ] = 12 + 5 = 17 (b) F(-6) = 3(-6) + 5 = -18 + 5 = -13 Example 2 Given that g ( x) x 2 3x 5 , evaluate g(-2). Solution g (2) (2) 2 3(2) 5 [Remember to always use brackets when you substitute] =4+6+5 = 15 Example 3 Given that f ( x) 4 x 3 and g ( x) 15 2 x , find (a) The value of a for which f(a) = 17 (b) Solve g(p) = 35 (c) Solve f(x) = g(x). Solution (a) Replace x by a in f(x) and equate to 17 4a – 3 = 17 4a = 20 a=5 (b) 15 – 2p = 35 -2p = 20 p = -10 (c) 4x 3 15 2x 4x 2x 15 3 6 x 18 x3 Page 22 LINEAR FUNCTIONS EXERCISE 1. Given that f ( x) 2 x 7 , find f(3). 2. Given that f ( x) 3x 10 , find f(-2). 3. Given that f ( x) 13 4 x , find f(4). 4. Given that f ( x) x 2 4 , find f(-5). 5. Given that g ( x) 2 x 2 3x 1 , find g(2). 6. Given that g ( x) 3x 2 4 x 5 , find g(-2). 7. Given that g ( x) (2 x 1) 2 , find g(-4). 8. Given that g ( x) 3( x 4) 2 5 , find g(5) . 9. Given f ( x) 10 2( x 3) 2 , find f(-2). 10. Given that f ( x) 2 x 2 5 , find f(3). Page 23 LINEAR FUNCTIONS 11. Given f ( x) 2 x 1 and g ( x) 10 3x , find 12. Given f ( x) 5 x 4 and g ( x) 18 2 x , find (a) The value of a for which f(a) = 13. (a) The value of a for which f(a) = 24. (b) Solve g(p) = 22 (b) Solve g(p) = 28 (c) Solve f(x) = g(x). (c) Solve f(x) = g(x). 13. Given f ( x) x 2 5 and g ( x) 5 x 1, find 14. Given that f ( x) x 2 3 and g ( x) 7 x 2 , find (a) The values of a for which f(a) = 20. (a) The values of a for which f(a) = 39. (b) Solve g(t) = 11 (b) Solve g(q) = 19. Page 24 LINEAR FUNCTIONS USING TECHNOLOGY TO SOLVE SIMULTANEOUS EQUATIONS To solve a pair of simultaneous equations, use the following steps: Main Keyboard 2D Insert the first equation in the first box, the second equation in the second box and x,y in the box outside. Exercise Solve the following equations using your calculator. (a) 𝑦 = 2𝑥 + 5 𝑎𝑛𝑑 𝑦 = 3𝑥 − 1 Answer : x = ……… and y = ………… (b) 𝑦 = 𝑥 + 4 𝑎𝑛𝑑 𝑦 = 4𝑥 − 8 Answer : x = ……… and y = ………… (c) 𝑥 + 𝑦 = 10 𝑎𝑛𝑑 2𝑥 − 𝑦 = 8 Answer : x = ……… and y = ………… (d) 3𝑥 − 𝑦 = 10 𝑎𝑛𝑑 2𝑥 + 5𝑦 = 1 Answer : x = ……… and y = ………… (e) 4𝑥 + 3𝑦 = 11 𝑎𝑛𝑑 5𝑥 − 𝑦 = 9 Answer : x = ……… and y = ………… (f) 𝑥 + 𝑦 = 9 𝑎𝑛𝑑 2𝑥 − 𝑦 = 6 Answer : x = ……… and y = ………… Page 25