Straight Line Graph

(3, 5) (3, 5) (3, 5) Coordinate pairs When we write a coordinate, for example, (3, 5) (3, 5) (3, 5) x-coordinate y-coordinate the first number is called the x-coordinate and the second number is called the y-coordinate. the first number is called the x-coordinate and the second number is called the y-coordinate. Link: S4 Coordinates and transformations 1 – coordinates. Together, the x-coordinate and the y-coordinate are called a coordinate pair.

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. x y All of the points lie on a straight line parallel to the y-axis. Stress that as long as the x-coordinate is 2 the y-coordinate can be any number: positive, negative or decimal. Encourage pupils to be imaginative in their choice of points that lie on this line. For example, (2, 6491637) (2, 43/78) or (2, –0.000003). Name five other points that will lie on this line. This line is called x = 2. x = 2

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). x y Stress that the graph of x = ‘something’ will always be parallel to the y-axis. In other words, it will always be vertical (not horizontal like the x-axis). For each graph shown in the example, ask pupils to tell you the coordinate of the point where the line cuts the x-axis. Ask pupils to tell you the equation of the line that coincides with the y-axis (x = 0). x = –10 x = –3 x = 4 x = 9

Graphs parallel to the x-axis What do these coordinate pairs have in common? (0, 1), (3, 1), (–2, 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. x y Stress that as long as the y-coordinate is 1 the x-coordinate can be any number: positive negative or decimal. Encourage pupils to be imaginative in their choice of points that lie on this line. For example, (1, 1934792) (1, 56/87) or (1, –0.0000047). y = 1 Name five other points that will lie on this line. This line is called y = 1.

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). x y y = 5 y = 3 Stress that the graph of y = ‘something’ will always be parallel to the x-axis. In other words, it will always be horizontal (not vertical like the y-axis). For each graph shown in the example, ask pupils to tell you the coordinate of the point where the line cuts the y-axis. Ask pupils to tell you the equation of the line that coincides with the x-axis (y = 0). y = –2 y = –5

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: Ask pupils if they can visualize the shape that the graph will have. This might be easier if they consider the points (0, 2), (1, 3), (2, 4) (3, 5) etc. Establish that the points will lie on a straight diagonal line. Stress that the graphs of all linear functions are straight lines. A function is linear if the variables are not raised to any power (other than 1). Ask pupils to suggest the coordinates of any other points that will lie on this line. Praise the most imaginative answers. y = x + 2 We can draw a graph of the function y = x + 2 by plotting points that obey this function.

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 y = x + 3 –3 –2 –1 1 2 3 Explain that when we construct a table of values, the value of y depends on the value of x. That means that we choose the values for x and substitute them into the equation to get the corresponding value for y. The minimum number of points needed to draw a straight line is two, however, it is best to plot several points to ensure that no mistakes have been made. The points given by the table can then be plotted to give the graph of the required function. 1 2 3 4 5 6 (–3, 0) (–2, 1) (–1, 2) (0, 3) (1, 4) (2, 5) (3, 6)

Drawing graphs of functions For example, y x to draw a graph of y = x – 2: y = x – 2 1) Complete a table of values: x y = x – 2 –3 –2 –1 1 2 3 –5 –4 –3 –2 –1 1 2) Plot the points on a coordinate grid. 3) Draw a line through the points. This slide summarizes the steps required to plot a graph using a table of values. 4) Label the line. 5) Check that other points on the line fit the rule.

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. Explain that the equation of a line can always be arranged to be in the form y = mx + c. It is often useful to have the equation of a line in this form because it tells us the gradient of the line and where it cuts the x-axis. These two facts alone can enable us to draw the line without have to set up a table of values. Ask pupils what they can deduce about two graphs that have the same value for m. Establish that if they have the same value for m, they will have the same gradient and will therefore be parallel. This is called the y-intercept and it has the coordinate (0, c). For example, the line y = 3x + 4 has a gradient of 3 and crosses the y-axis at the point (0, 4).

The gradient and the y-intercept Complete this table: equation gradient y-intercept y = 3x + 4 y = – 5 y = 2 – 3x 1 –2 3 (0, 4) x 2 1 2 (0, –5) (0, 2) –3 Complete this activity as a class exercise. (0, 0) y = x y = –2x – 7 (0, –7)

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 2y + 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 2y + x = 4 Explain that if the equation of a line is linear (that is if x and y are not raised to any power except 1), it can be arranged to be in the form y = mx + c. It is often useful to have the equation of a line in this form because it tells us the gradient of the line and where it cuts the y-axis. These two facts alone can enable us to draw the line without have to draw up a table of values. 2y = –x + 4 y = –x + 4 2 y = – x + 2 1 2

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 2y + 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 = – x + 2 1 2 1 2 – So the gradient of the line is and the y-intercept is 2.

What is the equation? What is the equation of the line passing through the points Look at this diagram: y C A B E G H F D 5 10 -5 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 x Ask pupils to give you the equations of the lines passing through the required points by considering the gradient and the y-intercept of each line. Ask pupils to tell you which lines are parallel. Ask how we can use the equations of the lines to find out which ones are parallel. Establish, that parallel lines have the same gradient and therefore, the x’s have the same coefficient. e) E and G y = 2 – x f) A and C? y = 10 – x

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 = 3m + 5 Subtracting 5: 6 = 3m Discuss ways to solve the problem. Some pupils may suggest plotting the point (3, 11) and drawing a straight line through this and the point (0, 5). The gradient of the resulting line will give the value for m. Ask pupils if they can suggest a method that does not involve drawing a graph. Establish that if the line passes through the point (3, 11) then we can substitute x = 3 and y = 11 into the equation y = mx + 5. Reveal the equation 11 = 3m + 5 on the board and talk through the steps leading to the solution of this equation. Dividing by 3: 2 = m m = 2 The equation of the line is therefore y = 2x + 5.

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 y x an upwards slope y x a horizontal line y x a downwards slope Positive gradient Zero gradient Negative gradient If a line is vertical its gradient cannot by specified.

Finding the gradient from two given points If we are given any two points (x1, y1) and (x2, y2) on a line we can calculate the gradient of the line as follows, y the gradient = change in y change in x x (x2, y2) y2 – y1 (x1, y1) Draw a right-angled triangle between the two points on the line as follows, x2 – x1 Formally define the gradient as the change in y/the change in x. Explain that since, for a straight line, the change in y is proportional to the corresponding change in x, the gradient will be the same no matter which two points we choose on a line. Explain how drawing a right-angled triangle on the line help us calculate its gradient. Explain too that since, for a straight line, the change in y is proportional to the corresponding change in x, the gradient will be the same no matter which two points we choose on a line. the gradient = y2 – y1 x2 – x1