KS 3 Mathematics A 5 Functions and graphs

KS 3 Mathematics A 5 Functions and graphs 1 of 51 © Boardworks Ltd 2004

A 5. 5 Graphs of functions 2 of 51 © Boardworks Ltd 2004

Coordinate pairs When we write a coordinate, for example, (3, 5) x-coordinate y-coordinate the first number is called the x-coordinate and the second number is called the y-coordinate. Together, the x-coordinate and the y-coordinate are called a coordinate pair. 3 of 51 © Boardworks Ltd 2004

Graphs parallel to the y-axis What do these coordinate pairs have in common? (2, 3), (2, 1), (2, – 2), (2, 4), (2, 0) and (2, – 3)? The x-coordinate in each pair is equal to 2. Look what happens when these points are plotted on a graph. All of the points lie on a straight line parallel to the y-axis. y x x=2 4 of 51 Name five other points that will lie on this line. This line is called x = 2. © Boardworks Ltd 2004

Graphs parallel to the y-axis All graphs of the form x = c, where c is any number, will be parallel to the y-axis and will cut the x-axis at the point (c, 0). y x x = – 10 5 of 51 x = – 3 x=4 x=9 © Boardworks Ltd 2004

Graphs parallel to the x-axis What do these coordinate pairs have in common? (0, 1), (4, 1), (– 2, 1), (1, 1) and (– 3, 1)? The y-coordinate in each pair is equal to 1. Look what happens when these points are plotted on a graph. All of the points lie on a straight line parallel to the x-axis. y y=1 x Name five other points that will lie on this line. This line is called y = 1. 6 of 51 © Boardworks Ltd 2004

Graphs parallel to the x-axis All graphs of the form y = c, where c is any number, will be parallel to the x-axis and will cut the y-axis at the point (0, c). y y=5 y=3 x y = – 2 y = – 5 7 of 51 © Boardworks Ltd 2004

Drawing graphs of functions The x-coordinate and the y-coordinate in a coordinate pair can be linked by a function. What do these coordinate pairs have in common? (1, 3), (4, 6), (– 2, 0), (0, 2), (– 1, 1) and (3. 5, 5. 5)? In each pair, the y-coordinate is 2 more than the x-coordinate. These coordinates are linked by the function: y=x+2 We can draw a graph of the function y = x + 2 by plotting points that obey this function. 8 of 51 © Boardworks Ltd 2004

Drawing graphs of functions Given a function, we can find coordinate points that obey the function by constructing a table of values. Suppose we want to plot points that obey the function y=x+3 We can use a table as follows: x – 3 – 2 – 1 0 1 2 3 y = x +3 0 1 2 3 4 5 6 (– 3, 0) (– 2, 1) (– 1, 2) (0, 3) (1, 4) (2, 5) (3, 6) 9 of 51 © Boardworks Ltd 2004

Drawing graphs of functions For example, to draw a graph of y = x – 2: y=x-2 1) Complete a table of values: x – 3 – 2 – 1 0 1 2 y = x – 2 – 5 – 4 – 3 – 2 – 1 0 3 1 2) Plot the points on a coordinate grid. 3) Draw a line through the points. 4) Label the line. 5) Check that other points on the line fit the rule. 10 of 51 © Boardworks Ltd 2004

Drawing graphs of functions 11 of 51 © Boardworks Ltd 2004

The equation of a straight line The general equation of a straight line can be written as: y = mx + c The value of m tells us the gradient of the line. The value of c tells us where the line crosses the y-axis. This is called the y-intercept and it has the coordinate (0, c). For example, the line y = 3 x + 4 has a gradient of 3 and crosses the y-axis at the point (0, 4). 12 of 51 © Boardworks Ltd 2004

Linear graphs with positive gradients 13 of 51 © Boardworks Ltd 2004

Investigating straight-line graphs 14 of 51 © Boardworks Ltd 2004

The gradient and the y-intercept Complete this table: 15 of 51 equation gradient y-intercept y = 3 x + 4 3 (0, 4) x y= – 5 2 1 2 (0, – 5) y = 2 – 3 x – 3 (0, 2) y=x 1 (0, 0) y = – 2 x – 7 – 2 (0, – 7) © Boardworks Ltd 2004

Rearranging equations into the form y = mx + c Sometimes the equation of a straight line graph is not given in the form y = mx + c. The equation of a straight line is 2 y + x = 4. Find the gradient and the y-intercept of the line. We can rearrange the equation by transforming both sides in the same way 2 y + x = 4 2 y = –x + 4 y= 2 y=– 1 x+2 2 16 of 51 © Boardworks Ltd 2004

Rearranging equations into the form y = mx + c Sometimes the equation of a straight line graph is not given in the form y = mx + c. The equation of a straight line is 2 y + x = 4. Find the gradient and the y-intercept of the line. Once the equation is in the form y = mx + c we can determine the value of the gradient and the y-intercept. y=– 1 x+2 2 So the gradient of the line is – 1 and the y-intercept is 2. 2 17 of 51 © Boardworks Ltd 2004

What is the equation? What is the equation of the line passing through the points Look at this diagram: 10 A G H 5 B F D -5 18 of 51 0 E C 5 10 a) A and E x=2 b) A and F y=x+6 c) B and E y=x– 2 d) C and D y=2 e) E and G y=2–x f) A and C? y = 10 – x © Boardworks Ltd 2004

Substituting values into equations A line with the equation y = mx + 5 passes through the point (3, 11). What is the value of m? To solve this problem we can substitute x = 3 and y = 11 into the equation y = mx + 5. This gives us: 11 = 3 m + 5 Subtracting 5: 6 = 3 m Dividing by 3: 2=m m=2 The equation of the line is therefore y = 2 x + 5. 19 of 51 © Boardworks Ltd 2004

Pairs 20 of 51 © Boardworks Ltd 2004

Matching statements 21 of 51 © Boardworks Ltd 2004

Exploring gradients 22 of 51 © Boardworks Ltd 2004

Gradients of straight-line graphs The gradient of a line is a measure of how steep a line is. The gradient of a straight line y = mx + c is given by change in y m= change in x For any two points on a straight line, (x 1, y 1) and (x 2, y 2) y 2 – y 1 m= x 2 – x 1 23 of 51 © Boardworks Ltd 2004

Contents A 8 Linear and real-life graphs A A 8. 1 Linear graphs A A 8. 2 Gradients and intercepts A A 8. 3 Parallel and perpendicular lines A A 8. 4 Interpreting real-life graphs A A 8. 5 Distance-time graphs A A 8. 6 Speed-time graphs 24 of 51 © Boardworks Ltd 2004

Coordinate pairs When we write a coordinate, for example, (3, 5) x-coordinate y-coordinate the first number is called the x-coordinate and the second number is the called y-coordinate. the y-coordinate. Together, the x-coordinate and the y-coordinate are called a coordinate pair. 25 of 51 © Boardworks Ltd 2004

Graphs parallel to the y-axis What do these coordinate pairs have in common? (2, 3), (2, 1), (2, – 2), (2, 4), (2, 0) and (2, – 3)? The x-coordinate in each pair is equal to 2. Look what happens when these points are plotted on a graph. All of the points lie on a straight line parallel to the y-axis. y x x=2 26 of 51 Name five other points that will lie on this line. This line is called x = 2. © Boardworks Ltd 2004

Graphs parallel to the y-axis All graphs of the form x = c, where c is any number, will be parallel to the y-axis and will cut the x-axis at the point (c, 0). y x x = – 10 27 of 51 x = – 3 x=4 x=9 © Boardworks Ltd 2004

Graphs parallel to the x-axis What do these coordinate pairs have in common? (0, 1), (4, 1), (– 2, 1), (1, 1) and (– 3, 1)? The y-coordinate in each pair is equal to 1. Look at what happens when these points are plotted on a graph. All of the points lie on a straight line parallel to the x-axis. y y=1 x Name five other points that will lie on this line. This line is called y = 1. 28 of 51 © Boardworks Ltd 2004

Graphs parallel to the x-axis All graphs of the form y = c, where c is any number, will be parallel to the x-axis and will cut the y-axis at the point (0, c). y y=5 y=3 x y = – 2 y = – 5 29 of 51 © Boardworks Ltd 2004

Plotting graphs of linear The x-coordinate and the y-coordinate in a coordinate functions pair can be linked by a function. What do these coordinate pairs have in common? (1, – 1), (4, 2), (– 2, – 4), (0, – 2), (– 1, – 3) and (3. 5, 1. 5)? In each pair, the y-coordinate is 2 less than the x-coordinate. These coordinates are linked by the function: y=x– 2 We can draw a graph of the function y = x – 2 by plotting points that obey this function. 30 of 51 © Boardworks Ltd 2004

Plotting graphs of linear Given a function, we can find coordinate points that obey functions the function by constructing a table of values. Suppose we want to plot points that obey the function y = 2 x + 5 We can use a table as follows: x – 3 – 2 – 1 0 1 2 3 y = 2 x + 5 – 1 1 3 5 7 9 11 (– 3, – 1) (– 2, 1) (– 1, 3) (0, 5) (1, 7) (2, 9) (3, 11) 31 of 51 © Boardworks Ltd 2004

Plotting graphs of linear y For example, functions to draw a graph of y = 2 x + 5: 1) Complete a table of values: x – 3 – 2 – 1 0 y = 2 x + 5 – 1 1 3 5 1 7 2 3 9 11 y = 2 x + 5 2) Plot the points on a coordinate grid. 3) Draw a line through the points. x 4) Label the line. 5) Check that other points on the line fit the rule. 32 of 51 © Boardworks Ltd 2004

Plotting graphs of linear functions 33 of 51 © Boardworks Ltd 2004

Contents A 8 Linear and real-life graphs A A 8. 1 Linear graphs A A 8. 2 Gradients and intercepts A A 8. 3 Parallel and perpendicular lines A A 8. 4 Interpreting real-life graphs A A 8. 5 Distance-time graphs A A 8. 6 Speed-time graphs 34 of 51 © Boardworks Ltd 2004

Gradients of straight-line graphs The gradient of a line is a measure of how steep the line is. The gradient of a line can be positive, negative or zero if, moving from left to right, we have an upwards slope y a horizontal line y y x x Positive gradient a downwards slope Zero gradient x Negative gradient If a line is vertical, its gradient cannot be specified. 35 of 51 © Boardworks Ltd 2004

Calculating gradients 36 of 51 © Boardworks Ltd 2004

Finding the gradient from two If we are given any two points (x , y ) and (x , y ) on a line we given points can calculate the gradient of the line as follows, 1 change in y the gradient = change in x Draw a right-angled triangle between the two points on the line as follows, y 2 – y 1 the gradient = x 2 – x 1 37 of 51 1 y 2 2 (x 2, y 2) y 2 – y 1 (x 1, y 1) x 2 – x 1 x © Boardworks Ltd 2004

Investigating linear graphs 38 of 51 © Boardworks Ltd 2004

The general equation of a straight line can be written as: straight line y = mx + c The value of m tells us the gradient of the line. The value of c tells us where the line crosses the y-axis. This is called the y-intercept and it has the coordinate (0, c). For example, the line y = 3 x + 4 has a gradient of 3 and crosses the y-axis at the point (0, 4). 39 of 51 © Boardworks Ltd 2004

The gradient and the y-intercept Complete this table: equation gradient y-intercept y = 3 x + 4 3 (0, 4) x – 5 2 1 2 (0, – 5) y = 2 – 3 x – 3 (0, 2) y=x 1 (0, 0) y = – 2 x – 7 – 2 (0, – 7) y= 40 of 51 © Boardworks Ltd 2004

Rearranging equations into the Sometimes the equation ofya = straight line graph is not given in form mx + c the form y = mx + c. The equation of a straight line is 2 y + x = 4. Find the gradient and the y-intercept of the line. Rearrange the equation by performing the same operations on both sides, 2 y + x = 4 subtract x from both sides: divide both sides by 2: 2 y = –x + 4 y= 2 y=– 1 x+2 2 41 of 51 © Boardworks Ltd 2004

Rearranging equations into the Sometimes the equation ofya = straight line graph is not given in form mx + c the form y = mx + c. The equation of a straight line is 2 y + x = 4. Find the gradient and the y-intercept of the line. Once the equation is in the form y = mx + c we can determine the value of the gradient and the y-intercept. y=– 1 x+2 2 So the gradient of the line is – 1 and the y-intercept is (0, 2). 2 42 of 51 © Boardworks Ltd 2004

Substituting values into A line with the equation y = mx + 5 equations passes through the point (3, 11). What is the value of m? To solve this problem we can substitute x = 3 and y = 11 into the equation y = mx + 5. This gives us, 11 = 3 m + 5 subtract 5 from both sides: 6 = 3 m divide both sides by 3: 2=m m=2 The equation of the line is therefore y = 2 x + 5. 43 of 51 © Boardworks Ltd 2004

What is the equation of the line? 44 of 51 © Boardworks Ltd 2004

Match the equations to the graphs 45 of 51 © Boardworks Ltd 2004

Contents A 8 Linear and real-life graphs A A 8. 1 Linear graphs A A 8. 2 Gradients and intercepts A A 8. 3 Parallel and perpendicular lines A A 8. 4 Interpreting real-life graphs A A 8. 5 Distance-time graphs A A 8. 6 Speed-time graphs 46 of 51 © Boardworks Ltd 2004

Investigating parallel lines 47 of 51 © Boardworks Ltd 2004

Parallel lines If two lines have the same gradient they are parallel. Show that the lines 2 y + 6 x = 1 and y = – 3 x + 4 are parallel. We can show this by rearranging the first equation so that it is in the form y = mx + c. 2 y + 6 x = 1 2 y = – 6 x + 1 subtract 6 x from both sides: – 6 x + 1 divide both sides by 2: y= 2 y = – 3 x + ½ The gradient m is – 3 for both lines, so they are parallel. 48 of 51 © Boardworks Ltd 2004

Matching parallel lines 49 of 51 © Boardworks Ltd 2004

Investigating perpendicular lines 50 of 51 © Boardworks Ltd 2004

Perpendicular lines If the gradients of two lines have a product of – 1 then they are perpendicular. In general, if the gradient of a line is m, then the gradient of – 1 the line perpendicular to it is. m Write down the equation of the line that is perpendicular to y = – 4 x + 3 and passes through the point (0, – 5). The gradient of the line y = – 4 x + 3 is – 4. 1 The gradient of the line perpendicular to it is therefore 4. 1 The equation of the line with gradient and y-intercept – 5 is, y= 51 of 51 1 4 4 x– 5 © Boardworks Ltd 2004

Matching perpendicular lines 52 of 51 © Boardworks Ltd 2004