AQA Level 2 Further mathematics Number & algebra. Section 3: Functions and their graphs

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1 AQA Level Further mathematics Number & algebra Section : Functions and their graphs Notes and Eamples These notes contain subsections on: The language of functions Gradients The equation of a straight line The intersection of two lines Sketching the graphs of functions The language of functions Start with an positive number. Multipl b, and then add. The rule above is an eample of a function. A function is a rule which can be applied to an object or number in a particular set, and outputs a new object or number. When ou are working with numbers, a function can usuall be epressed algebraicall. The function above can be written like this: f() = + > 0 The set of possible inputs for the function is called the domain of the function. In this case, the domain is the set of all positive numbers, so this can be written as > 0. The set of possible outputs for the function is called the range of the function. In this case, because the domain is > 0, the output must be greater, so the range is f() >. If we had chosen a different domain, the range would have been different. For eample, we could have chosen all real numbers for the domain, in which case the range would also be all real numbers. Eample The function f() is defined as f() = ( )² + > 0 (i) Find the value of f(5). (ii) What is the range of f()? (iii) Find an epression for f( + ). Give our answer in its simplest form. (i) f(5) = (5 )² + = 4² + = 6 + = 9. (ii) The smallest possible value of f() is (when = ). So the range of f() is given b f() >. The epression in the bracket cannot be negative its smallest possible value is 0 of /06/ MEI

2 AQA FM Number & algebra Notes and Eamples (iii) f( + ) ( ) ( ) 4 Replace b + in the definition of the function Eample The function f() is defined as f ( ). (i) Find f(4). (ii) f() has domain all ecept = a. What is the value of a? (i) 4 4 f (4) (ii) a = - When = -, the denominator is zero so the function isn t defined. So = - has to be ecluded from the domain. Gradients You will have probabl have met gradients before. To revise finding the gradient of a line from a diagram, use the interactive questions The gradient of a line. Remember that lines which go downhill have negative gradients. To find the gradient of a straight line between two points formula gradient. Eample P is the point (-, ). Q is the point (5, ). Calculate the gradient of PQ and,,, use the Choose P as (, ) and Q as (, ). or vice versa: it will still give the same answer (WHY?) Gradient of PQ = = 5 ( ) = 6 = 4 Notes: () Draw a sketch and check that our answer is sensible (e.g. has negative gradient). () Check that ou get the same result when ou choose Q as ( For more eamples on gradient, look at, ) and P as ( the Flash resource, ). Gradient of a line. You ma also find the Mathcentre video The gradient of a straight line segment useful. of /06/ MEI

3 AQA FM Number & algebra Notes and Eamples For further practice in eamples like the one above, tr the interactive questions The gradient of a line between two points and Collinear points. The equation of a straight line The equation of a straight line is often written in the form = m + c, where m is the gradient and c is the intercept with the -ais. Eample 4 Find (i) the gradient and (ii) the -intercept of the following straight-line equations. (a) 5 (b) 0 (a) Rearrange the equation into the form = m + c. 5 becomes so m = and 5 c 5 (i) The gradient is 5 (ii) The -intercept is Note the minus sign (b) Rearrange the equation into the form = m + c. 0 becomes giving = + so m = and c = (i) The gradient is (ii) The -intercept is. Note the minus sign Sometimes ou ma need to sketch the graph of a line. A sketch is a simple diagram showing the line in relation to the origin. It should also show the coordinates of the points where it cuts one or both aes. You can eplore straight line graphs using the Flash resources Equation of a line = m + c and Equation of a line a + b + c = 0. You ma also find the Mathcentre video Equations of a straight line and Linear functions and graphs useful. Eample 5 Sketch the lines (a) 5 (b) 0 of /06/ MEI

4 AQA FM Number & algebra Notes and Eamples (a) 5 When = 0, 5 When = 0, 5 (b) 0 When = 0, When = 0, Sometimes ou ma need to find the equation of a line given certain information about it. If ou are given the gradient and intercept, this is eas: ou can simpl use the form = m + c. However, more often ou will be given the information in a different form, such as the gradient of the line and the coordinates of one point on the line (as in Eample 6) or just the coordinates of two points on the line (as in Eample ). In such cases ou can use the alternative form of the equation of a straight line. For a line with gradient m passing through the point,, the equation of the line is given b m. Eample 6 Find the equation of the line with gradient and passing through (, -). m = and (, ) 4 of is (, /06/ -) MEI

5 AQA FM Number & algebra Notes and Eamples The equation of the line is m 6 You should check that the point (, -) satisfies our line. If it doesn t, ou must have made a mistake! You can see more eamples like this using the Flash resource Equation of a line =m( ). In the net eample, ou are given the coordinates of two points on the line. Eample P is the point (, ). Q is the point (-, 5). Find the equation of PQ. Choose P as (, ) and Q as (, ). One method is to find the gradient and then use this value and one of the points in m Gradient of PQ = 5 = = 4 = 4 Now use m You should check that P and Q satisf our line. An alternative approach to the above eamples is to put the formula for m into the straight line equation to obtain and then make the substitutions. This is equivalent to the first method, but does not involve calculating m separatel first. For further practice in eamples like the one above, tr the interactive questions The equation of a line between two points. 5 of /06/ MEI

6 AQA FM Number & algebra Notes and Eamples The Lines Activities include a variet of activities which involve the equations of lines and their properties. The intersection of two lines The point of intersection of two lines is found b solving the equations of the lines simultaneousl. This can be done in a variet of was. When both equations are given in the form = then equating the right hand sides is a good approach (see below). If both equations are not in this form, ou can re-arrange them into this form first, then appl the same method. Alternativel, ou can use the elimination method if the equations are in an appropriate form. Eample Find the point of intersection of the lines and Substitute into one of the equations to find Substituting = into gives The point of intersection is (, ) Check that (, ) satisfies the second equation. You can see more eamples like this using the Flash resource Intersection of two lines. Sketching the graphs of functions You can sketch the graph of a linear or quadratic function b thinking about where the graph cuts the coordinate aes. A linear function is of the form f() = a + b. The graph of = f() is a straight line. You can find where it cuts the -ais b substituting = 0, and ou can find where it cuts the -ais b substituting = 0, as in Eample 5. A quadratic function is of the form f() = a² + b + c. The graph of = f() is a curve called a parabola. You can find where the graph cuts the - ais b substituting = 0. You can find where it cuts the -ais b solving the equation a² + b + c = 0. If the equation has no solutions, then the graph does not cut the -ais. 6 of /06/ MEI

7 AQA FM Number & algebra Notes and Eamples Eample 9 Sketch the graph of = f() for each of the following functions: (i) f() = (ii) f() = ² (iii) f() = ² + (i) This is a straight line graph. f(0) = - so the graph crosses the -ais at (0, -) When = 0, = 0 so, so the graph crosses the -ais at,0. - (ii) f(0) = -, so the graph crosses the -ais at (0, -) When = 0, 0 ( )( ) 0 or so the graph crosses the -ais at (, 0) and (-, 0) - - (iii) f(0) =, so the graph crosses the -ais at (0, ) When = 0, ² + = 0 has no solutions, so the graph does not cut the -ais. Functions are sometimes defined piecewise this means that the domain is split into separate parts and the function is defined differentl for each part. Here is an eample. of /06/ MEI

8 AQA FM Number & algebra Notes and Eamples Eample 0 A function f() is defined as f ( ) 0 0 Sketch the graph of = f() 4 At =, both + and have the value 4 of /06/ MEI