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Learn to plot, solve, and transform nonlinear function graphs in mathematics with practical examples and step-by-step instructions.
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KS4 Mathematics A9 Graphs of non-linear functions
A9 Graphs of non-linear functions Contents • A A9.2 Graphs of important non-linear functions • A A9.3 Using graphs to solve equations • A A9.1 Plotting curved graphs A9.4 Solving equations by trial and improvement • A A9.5 Function notation • A A9.6 Transforming graphs • A
Functions × 3 – 1 a function machine, a mapping arrow, x 3x – 1 x y an equation, or function notation. y =3x – 1 f(x) =3x – 1 In maths, what do we mean by a function? In maths, a function is a rule that maps one number called the input (x) onto an other number, the output (y). There are many ways of expressing a function. For example, the function “multiply by 3 and subtract 1” can be written using:
Linear and non-linear functions The simplest type of function is a linear function. The equation of a linear function can always be arranged in the form y = mx + c, where m and c are constants. The graph of a linear function will always be a straight line. If a function cannot be arranged in the form y = mx + c then it is a non-linear function. The graph of a non-linear function is usually curved.
Non-linear functions y = x2 + 1 y = 7x3 – 3x y = 2x+x8 5 y = 3 + 2x y = – 6 x – 2 Examples of non-linear functions include, We can plot the graphs of non-linear functions using a graphics calculator or a computer. We can also use a table of values.
Using a table of values x –3 –2 –1 0 1 2 3 y = x2 – 3 Plot the graph of y = x2 – 3 for values of x between –3 and 3. We can use a table of values to generate coordinates that lie on the graph as follows: 6 1 –2 –3 –2 1 6 (–3, 6) (–2, 1) (–1, –2) (0, –3) (1, –2) (2, 1) (3, 6)
Using a table of values x –3 –2 –1 0 1 2 3 y = x2 – 3 6 1 –2 –3 –2 1 6 y y 5 4 3 2 1 x –3 –2 –1 0 1 2 3 –1 –2 The points given in the table are plotted … … and the points are then joined together with a smooth curve. The shape of this graph is called a parabola. It is characteristic of a quadratic function.
Using a table of values Plot the graph of y = x3 – 7x + 2 for values of x between –3 and 3. x –3 –2 –1 0 1 2 3 x3 – 7x + 2 y = x3 – 7x + 2 This function is more complex and so it is helpful to include more rows in the table to show each stage in the substitution. –27 –8 –1 0 1 8 27 + 21 +14 + 7 + 0 – 7 – 14 – 21 + 2 + 2 + 2 + 2 + 2 + 2 + 2 –4 8 8 2 –4 –4 8
Using a table of values x –3 –2 –1 0 1 2 3 y = x3 – 7x + 2 –4 8 8 2 –4 –4 8 y y 10 8 6 4 2 x –3 –2 –1 0 1 2 3 –2 –4 The points given in the table are plotted … … and the points are then joined together with a smooth curve. The shape of this graph is characteristic of a cubic function.
A9 Graphs of non-linear functions Contents A9.1 Plotting curved graphs • A • A A9.3 Using graphs to solve equations • A A9.2 Graphs of important non-linear functions A9.4 Solving equations by trial and improvement • A A9.5 Function notation • A A9.6 Transforming graphs • A
Quadratic functions y = –3x2 y = x2 y = x2 – 3x A quadratic function always contains a term in x2. It can also contain terms in x or a constant. Here are examples of three quadratic functions: The characteristic shape of a quadratic function is called a parabola.
Cubic functions y = x3– 4x y = x3+ 2x2 A cubic function always contains a term in x3. It can also contain terms in x2 or x or a constant. Here are examples of three cubic functions: y = -3x2– x3
Reciprocal functions y = y = y = –4 3 x x 1 x A reciprocal function always contains a fraction with a term in x in the denominator. Here are examples of three simple reciprocal functions: In each of these examples the axes form asymptotes. The curve never touches these lines.
Exponential functions y = 2x y = 3x y = 0.25x An exponential function is a function in the form y = ax, where a is a positive constant. Here are examples of three exponential functions: In each of these examples, the x-axis forms an asymptote.
The equation of a circle y r y x 0 x One more graph that you should recognize is the graph of a circle centred on the origin. We can find the relationship between the x and y-coordinates on this graph using Pythagoras’ theorem. (x, y) Let’s call the radius of the circle r. We can form a right angled triangle with length y, height x and radius r for any point on the circle. Using Pythagoras’ theorem this gives us the equation of the circle as: x2 + y2 = r2
A9 Graphs of non-linear functions Contents A9.1 Plotting curved graphs • A A9.2 Graphs of important non-linear functions • A • A A9.3 Using graphs to solve equations A9.4 Solving equations by trial and improvement • A A9.5 Function notation • A A9.6 Transforming graphs • A
Using graphs to solve equations Solve the equation 2x2 – 5 = 3x using graphs. We can do this by considering the left-hand side and the right-hand side of the equation as two separate functions. 2x2 – 5 = 3x y = 2x2 – 5 y = 3x The points where these two functions intersect will give us the solutions to the equations.
Using graphs to solve equations 10 8 6 4 2 –4 –3 –2 –1 0 1 2 3 4 –2 –4 –6 The graphs of y = 2x2 – 5 and y = 3xintersect at the points: y = 2x2 – 5 y = 3x (2.5, 7.5) (–1, –3) and (2.5, 7.5). The x-value of these coordinates give us the solution to the equation 2x2 – 5 = 3x as (–1,–3) x = –1 and x = 2.5
Using graphs to solve equations Solve the equation 2x2 – 5 = 3x using graphs. Alternatively, we can rearrange the equation so that all the terms are on the right-hand side, 2x2 – 3x–5 = 0 y = 2x2 – 3x–5 y = 0 The line y = 0 is the x-axis. This means that the solutions to the equation 2x2 – 3x– 5 = 0 can be found where the function y = 2x2 – 3x– 5intersects with the x-axis.
Using graphs to solve equations 10 8 6 4 2 –4 –3 –2 –1 0 1 2 3 4 –2 –4 –6 The graphs of y = 2x2– 3x– 5 and y = 0intersect at the points: y = 2x2 – 3x – 5 (–1, 0) and (2.5, 0). The x-value of these coordinates give us the same solutions (2.5, 0) (–1,0) y = 0 x = –1 and x = 2.5
Using graphs to solve equations Solve the equation x3 – 3x = 1using graphs. This equation does not have any exact solutions and so the graph can only be used to find approximate solutions. A cubic equation can have up to three solutions and so the graph can also tell us how many solutions there are. Again, we can consider the left-hand side and the right-hand side of the equation as two separate functions and find the x-coordinates of their points of intersection. x3 – 3x = 1 y = x3 – 3x y = 1
Using graphs to solve equations 10 8 6 4 2 –4 –3 –2 –1 0 1 2 3 4 –2 –4 –6 y = x3 – 3x The graphs of y = x3 – 3x and y = 1intersect at three points: This means that the equation x3 – 3x = 1 has three solutions. y = 1 Using the graph these solutions are approximately: x = –1.5 x = –0.3 x = 1.9
A9 Graphs of non-linear functions Contents A9.1 Plotting curved graphs • A A9.2 Graphs of important non-linear functions • A A9.3 Using graphs to solve equations • A A9.4 Solving equations by trial and improvement • A A9.5 Function notation • A A9.6 Transforming graphs • A
Solving equations by trial and improvement The equation x3 – 3x = 1has a solution when x is approximately equal to 1.9. Find this solution to 3 decimal places. The value 1.9 was found using a graph. We can improve the accuracy of this answer by substituting 1.9 into the equation and noting whether it is too high or too low. We then substitute a better value and continue the process until we have a solution to the required degree of accuracy. This method of finding a solution is called trial and improvement.
Solving equations by trial and improvement x x3 – 3x comment 1.9 1.8 1.85 1.87 1.88 The equation x3 – 3x = 1has a solution when x is approximately equal to 1.9. Find this solution to 3 decimal places. Set up a table as follows, 1.159 too high 0.432 too low This tells us that the solution is between 1.8 and 1.9 0.781625 too low 0.929203 too low 1.004672 too high
Solving equations by trial and improvement x x3 – 3x comment 1.875 1.878 1.879 1.8795 The equation x3 – 3x = 1has a solution when x is approximately equal to 1.9. Find this solution to 3 decimal places. The solution is between 1.87 and 1.88, so try 1.875 next, 0.9667969 too low 0.9894882 too low 0.9970744 too low The solution is between 1.879 and 1.880. 1.0008718 too high The solution is 1.879 to 3 decimal places.
A9 Graphs of non-linear functions Contents A9.1 Plotting curved graphs • A A9.2 Graphs of important non-linear functions • A A9.3 Using graphs to solve equations • A A9.5 Function notation A9.4 Solving equations by trial and improvement • A • A A9.6 Transforming graphs • A
Functions a mapping arrow, a function machine, or x x2 square x y an equation, y =x2 Remember, a function is a rule that maps one number called the input (x) onto an other number, the output (y). For example, the function “square” can be written using, One more way of expressing a function is to use function notation. For example, f(x) =x2 “f of x equals x squared”
Function notation We write f(x) =x2 to define the function f. The function f can then act on any number, term or expression that is in the brackets. For example, f(5) =52= 25 f(–2) = (–2)2= 4 f(a) =a2 f(x + 4) =(x + 4)2 = x2 + 8x + 16 f(–x) =(–x)2 = x2
Function notation Suppose g(x) = 2x – 5 Find g(4) = 2 × 4– 5 = 3 g(1.5) = 2 × 1.5– 5 = –2 g(a) = 2a – 5 g(x + 3) = 2(x + 3) – 5 = 2x + 6 – 5 = 2x + 1 g(x)+ 3 = 2x – 5 + 3 = 2x – 2 g(2x) = 2 × 2x – 5= 4x– 5 2g(x) = 2(2x – 5)= 4x– 10
A9 Graphs of non-linear functions Contents A9.1 Plotting curved graphs • A A9.2 Graphs of important non-linear functions • A A9.3 Using graphs to solve equations • A A9.6 Transforming graphs A9.4 Solving equations by trial and improvement • A A9.5 Function notation • A • A
Transforming graphs of functions Graphs can be transformed by translating, reflecting, stretching or rotating them. The equation of the transformed graph will be related to the equation of the original graph. When investigating transformations it is most useful to express functions using function notation. For example, suppose we wish to investigate transformations of the function f(x) = x2. The equation of the graph of y = x2, can be written as y = f(x).
Vertical translations x The graph of y = f(x) + a is the graph of y = f(x) translated by the vector . 0 a Here is the graph of y = x2, where y = f(x). This is the graph of y = f(x) + 1 y and this is the graph of y = f(x) + 4. What do you notice? This is the graph of y = f(x) – 3 and this is the graph of y = f(x) – 7. What do you notice?
Horizontal translations x The graph of y = f(x + a) is the graph of y = f(x) translated by the vector . –a 0 Here is the graph of y = x2 – 3, where y = f(x). This is the graph of y = f(x – 1), y and this is the graph of y = f(x – 4). What do you notice? This is the graph of y = f(x + 2), and this is the graph of y = f(x + 3). What do you notice?
Reflections in the x-axis x The graph of y = –f(x) is the graph of y = f(x) reflected in the x-axis. Here is the graph of y = x2 –2x – 2, where y = f(x). y This is the graph of y = –f(x). What do you notice?
Reflections in the y-axis x The graph of y = f(–x) is the graph of y = f(x) reflected in the y-axis. Here is the graph of y = x3 + 4x2 – 3 where y = f(x). y This is the graph of y = f(–x). What do you notice?
Stretches in the y-direction The graph of y = af(x) is the graph of y = f(x) stretched parallel to the y-axis by scale factor a. Here is the graph of y = x2, where y = f(x). This is the graph of y = 2f(x). y What do you notice? This graph is is produced by doubling the y-coordinate of every point on the original graph y = f(x). This has the effect of stretching the graph in the vertical direction. x
Stretches in the x-direction x The graph of y = f(ax) is the graph of y = f(x) stretched parallel to the x-axis by scale factor . 1 a Here is the graph of y = x2 + 3x – 4, where y = f(x). This is the graph of y = f(2x). y What do you notice? This graph is is produced by halving the x-coordinate of every point on the original graph y = f(x). This has the effect of compressing the graph in the horizontal direction.
