Revision Topic 6: Mathematical models

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1 Revision Topic 6: Mathematical models Chapter 17: Functions and graphs What is a function? A function is a machine that does something to one number to turn it into another. Starting with a set of input numbers, the function will create a second set of output numbers. Functions are often written as f( x ) or gx, ( ) but you can use any letters for the function name and the variable. Many common functions can be expressed using a formula, which may involve specific elementary functions such as sin( x ) or e x. You are expected to use your GDC when working with functions and graphs. You need to know the following vocabulary for functions: Domain Range Graph Mapping diagram The set of input values that go into the function The set of output values that are created by the function from the input values The visual diagram formed by plotting pairs of coordinates where the first number is from the domain and the second number is the corresponding value from the range An alternative diagram that shows the links between numbers in the domain and numbers in the range. For example: Copyright Cambridge University Press All rights reserved. Page 1 of 9

2 Finding the range when given the function and the domain To do this, use your GDC as follows: Texas TI-84 Enter the function into the GDC. Casio fx-9750gii Access the values in the table and scroll down to look at the Y1 values corresponding to each X value. The X values are the domain and the Y1 values give the range. Finding the domain and range from a graph You need to be able to work out the domain and range of a function from its graph. Remember the following points: The domain is made up of the values of x for which f( x ) has output values (i.e. y- coordinates). The range is found by putting every value of the domain into f( x ) in place of the x and 3 working out the answer. For example, if f( x) = x 3x+ 5, then for x =4 you would calculate 3 f (4) = = 57. Copyright Cambridge University Press All rights reserved. Page 2 of 9

3 For either the domain or range, there are four possible situations: Situation Example of how the graph might look like In formal notation this would look like The values are unlimited The domain is x Ρ There is a minimum value The domain is x 2 There is a maximum value The range is f( x) 5 The values are between two limits The domain is 0 x 7 The domain and range could each be in either one of these categories and should be considered separately when analysing the graph. Copyright Cambridge University Press All rights reserved. Page 3 of 9

4 Rational functions These are functions that are in the form of a fraction or ratio. If the denominator of the fraction contains the variable x of the domain, then the graph will have a break in it. A line which shows where the break occurs is called an asymptote. This graph has two asymptotes: a horizontal one with equation y = 2 and a vertical one with equation x = 3. If you use your GDC to plot the graph of a rational function, you can find the values for the asymptotes by looking at the table of values: Vertical asymptote Horizontal asymptote Look for where there is an ERROR message in the second (Y1) column and use the value of X that corresponds to that error. The values in the second (Y1) column will get closer and closer to a certain number. This number is the value of y to use for the equation. Drawing graphs and diagrams In the exam and when solving other problems, you often need to create an appropriate diagram. When asked to draw, plot or sketch, you need to know what to do: Wording Sketch Plot Draw This means you will need to do the following Use your GDC to draw the graph, and then make a diagram based on it that shows the key features of the graph such as intercepts, asymptotes and turning points. Calculate a set of coordinate pairs, plot these points, and then draw a line or curve through them. Make a diagram that is accurate and to scale; straight lines should be drawn with a ruler. Solving equations using graphs on your GDC Split the equation at the equals sign and put each half in as a graph equation on your GDC. Draw the two graphs; the solution is the x value(s) where the lines cross. Use the intersection tool to find the solution. If the lines don t cross, then there is no solution. 3x 2 2 For example, if the equation to solve is 4e + 6 = 7x 5x + 2, then split it to make 3x 2 y = 4e + 6 and y x x 2 = , draw the graphs of these two equations and look for their intersection. Copyright Cambridge University Press All rights reserved. Page 4 of 9

5 Chapter 18: Linear and quadratic models Linear models Linear means straight line, so linear equations will be of the form y = mx + c or ax + by + d = 0 Linear functions are of the form f ( x) = mx + c When solving problems involving linear models, it is helpful to draw a graph of the equations. The answer is normally where the lines intersect with each other or with one of the axes. Commonly asked questions can involve profit = revenue cost Quadratic functions and their graphs Questions may be stated using either the y = or the f( x ) = notation, and it is important that you use the correct format in written answers, even if you can readily interchange the two forms in your mind. A quadratic function has the form = + + where a 0. 2 f ( x) ax bx c The graph of a quadratic function is a parabola or U-shaped curve with the following properties: The curve is symmetrical about the vertical line through its minimum or maximum point. If a > 0, the curve will be like a ; if a < 0, the curve will be upside down and so look like. You need to be able to find certain properties of the curve and relate them to the quadratic function or vice versa. To find How to work this out from the function How to get this from the curve x-intercept Solve the equation formed by setting f( x ) = 0 Use the root function on your GDC y-intercept Calculate f(0) = c Look at the value of y when x = 0 Pick two points on a horizontal line that The equation b The equation is x = intersects the curve. of the line of 2a The equation of the line will be symmetry (this is given in the formula booklet) x= average of their x-coordinates The value of b Use the x value of the symmetry line and the value of a to get b (= 2ax). The minimum or maximum value of f( x ) Find the value of f b 2a Use the max or min function on your GDC Copyright Cambridge University Press All rights reserved. Page 5 of 9

6 You also need to be able to define the following values in relation to the function: Domain The x values for which f( x ) has an output (y) value Range The possible values of f( x ) Quadratic models In problems involving quadratic models, you will be given a function in the form described above. To find the answers to any questions asked, you should do the following: Draw the function on your GDC. Interpret the text in the question and write it down in mathematical notation. For example, if the question involves time, then at the beginning t = 0 ; and if the process takes ten seconds, then you have t = 10 at the end. When writing down your answer, put it in the context of the original question. Copyright Cambridge University Press All rights reserved. Page 6 of 9

7 Chapter 19: Exponential and polynomial functions Exponential functions and their graphs An exponential function is a function where the output value is multiplied by a fixed amount for each unit of increase in the input value. x An exponential growth function has the form f ( x) = ka + c where a > 1 and k 0. x An exponential decay function has the form f ( x) = ka + c (with a negative sign in the exponent). x x The graphs of f ( x) = ka + c and f ( x) = ka + c are reflections of each other in the y-axis. The y-intercept value is k + c. The value of c affects the graph s vertical position. If c > 0 the curve is shifted up relative to x the y = ka curve; if c < 0 the curve is shifted down. If k > 0 the curve lies above y = c; if k < 0 the curve lies below y = c. The size of k determines how steep the curve is. Horizontal asymptote Exponential graphs level off to a particular value, which they get close to but never actually reach. This value corresponds to a horizontal asymptote. x The equation of the asymptote will be y = c. If the function is of the form f ( x) = ka, with no c, then the horizontal asymptote is y = 0. Copyright Cambridge University Press All rights reserved. Page 7 of 9

8 Polynomial functions A polynomial function is made up of powers of the independent variable, normally x. It can be written in the following form: n m f ( x) = ax + bx + ; n, m Ν The powers must be non-negative integers, and there may be more than two terms. Here are some elements of the graphs of polynomial functions that you may need to identify, either to draw or sketch the curve or to answer a question. Graph element x-intercept(s) y-intercept Turning points What it means in terms of the function The root(s) or solution(s) of the equation f( x ) = 0 The value of f (0) The maximum or minimum points of f( x ) How to find it Use the GDC to solve f( x ) = 0 using the graph or the equation solver Use the GDC table to find this value, or read it off from the function (it will be the constant term, i.e. a number on its own) Use the GDC graph and the minimum and maximum functions Illustration Rational functions and vertical asymptotes n m In a function of the form f ( x) = ax + bx +, if any of the powers is a negative integer, then the function is not a polynomial; instead, it is a rational function, and includes a fraction which has the variable in the denominator. This means the graph will have a vertical asymptote. You can find the x value of the vertical asymptote by looking at the table on your GDC; find the value in the first column that gives an ERROR message in the second column. Copyright Cambridge University Press All rights reserved. Page 8 of 9

9 Exponential and polynomial models When working with an exponential or polynomial model, the formula may seem very complex, but you are expected to use your GDC to help you find the answer. Before you start trying to answer any questions: Texas TI-84 Put the model into your GDC. Do this in the same way as you would enter any equation; it might mean that you have to switch to using X instead of the variable given in the formula. This doesn t matter just remember to switch back to the original variable when you write down your final answer. Change the domain to fit the model you are working with: Set the minimum value of the domain and the amount that you want it to increment by. Casio fx-9750gii Then use the table to answer questions relating to particular values of either the domain or the range. Copyright Cambridge University Press All rights reserved. Page 9 of 9