Graphs and relations Construction and interpretation of graphs

If a relationship exists between variables, one can be said to be the function of the other. You can describe a function with a table, a rule, or a graph. A function whose graph is a straight line is called a linear function. The general equation of a straight line is where m is the gradient and c is the y-intercept. (Draw graphs)

Example, Ex 10A, Q.1, 4 You do Ex 10A, Q. 5, 6

Sketching straight-line graphs The graph of a straight line can be obtained by plotting and joining together any two points on the line. The three most common ways of sketching a straight-line graph by hand are: – Gradient-intercept method. – x- and y- intercept method. – Sketching a line over a required interval

Gradient-intercept form Usually easiest when equation is in form. 1.Plot the y-intercept 2.The gradient is given by m = rise/run. Write the gradient as a fraction and identify the rise and the run. 3.Plot the second point by starting from the y- intercept and moving across and up or down as suggested by the gradient. 4.Join the two points together with a straight line and label the graph. Example, Ex 10A, Q.7b

x- and y- intercept method Usually easiest when equation is in form. 1.If a point is on the y-axis, its x-coordinate is 0. To find the y-intercept, substitute 0 for x and solve the equation. 2.If a point is on the x-axis, its y-coordinate is 0. To find the x-intercept, substitute 0 for y and solve the equation. 3.Plot the x- and y- intercepts on a set of axes. 4.Join the two points together with a straight line and label the graph. Example, Ex 10A, Q.7a

Sketching a line over a required interval To sketch a graph between two given x-values, its end points must be known. We obtain the coordinates of the end points and join them together. 1.Rearrange the equation to make y the subject. 2.Substitute each of the two given x-values and find the corresponding values of y. 3.Plot the end points on a set of axes. 4.Join the two points together and label the graph. Example, Ex 10A, Q.7g

You do Ex 10A, Q. 7c, d, f, h

Applications of straight-line graphs When modelling a linear relationship, remember that the y-intercept represents the value of the function when x = 0. This is usually the original value of something or a fixed cost. The gradient represents the rate of change of the y-value with respect to the x-value. This is the change in y as x increases by 1 unit.

Example, Ex 10A, Q. 9 You do Ex 10A, Q. 10, 11

Extrapolation and interpolation Extrapolation examines the relationship between two variables beyond the limit of the data. Interpolation is to infer the relationship between two distinct data points.

Example, Ex 10A, Q.12 You do Ex 10A, Q. 13, 15, 16