Functions and Their graphs Mrs. Aldous, Mr. Beetz & Mr. Thauvette IB DP SL Mathematics

You should be able to… Use the features of your GDC to graph a variety of functions Use your GDC to investigate key features of graphs, including maximum values, minimum values, points of intersections, and zeros Find a value of a function Find the range of a functions on a given domain Identify horizontal and vertical asymptotes in functions where appropriate Solve equations using a graphical approach

You should know… A function is usually denoted by a small letter of the alphabet, such as f or g; the notation f(x) or g(x) is the value of the function at x The domain of a function is the set of x-values for which a function is defined The range of a function is best found by looking at the maximum and minimum values of the graph of the function on a given domain

Basic Functions

Substituting numbers into functions A function can be written as: Try some of these: Substituting is replacing the x so that, a) f(7)= 4 b) f(-2)= 1 Check these mentally: 4x3-3=9 a) g(3)= 2 b) g(-1)= 6 4x0-3=-3 a) h(1)= 27 b) h(-5)= 3 4x-2-3=-11

Solving functions f(x)=4 x=7 g(x)=0 x=1 and x=2 x=-0.87 and x=-5.121 Solving a function involves finding a solution for x. Try some of these: Solve f(x)=0 This is a simple algebra problem: f(x)=4 x=7 g(x)=0 x=1 and x=2 Check f(x)=17 x=-0.87 and x=-5.121 x=5 h(x)=4

Function vs. Mapping Notation

Drawing a graph and adjusting the window Enter: Y1 = x – 3 Now take a look at the graph in “standard” view: To pick another window to view the graph: Now enter your own values and see how it affects the viewing window when you look at the graph.

Solving Points on a Graph Using the graph: Y1 = x – 3 We can solve the following using the CALC menu on the GDC:

More solving points on a graph Now for a quadratic: 1. Enter this quadratic and get a window so that you can see all the information needed. 2. Find both roots using CALC > 2:zero. You will need to give a left and right bound and a guess. x=-4 and x=7 5. Find f(4) using algebra or CALC > 1:value. Have students try y = x^2 + 3x – 9. 3. Find the coordinates of the minimum using CALC > 3:minimum. y=-24 6. Find f(x)=5 using algebra or CALC > 5:intersect. 4. Find the y-intercept using CALC > 1:value. x=-4.44 y=-28

Graphs of Functions The x-intercepts of a function are the values of x for which y = 0. They are the zeros (i.e., solutions, roots) of the function. The y-intercept of a function is the value of y when x = 0. Beginning course details and/or books/materials needed for a class/project.

Graphs of Functions An asymptote is a line that the graph approaches or begins to look like as it tends to infinity in a particular direction. x = 2 horizontal asymptote vertical asymptote y = 2 A schedule design for optional periods of time/objectives.

Graphs of Functions To find vertical asymptotes, look for values of x for which the function is undefined: If find where If find where Objectives for instruction and expected results and/or skills developed from learning. To find horizontal asymptotes, consider the behaviour as

Asymptotes An asymptote is a line that a graph gets closer to but never reaches. This part of the functions is never 0, so y=-3 is never possible. What is not available for the domain? What number is not in the range? x=2 y=-3 The asymptotes are x=2 and y=-3. Graph x=2.

Now try one on your own 1. Use your GDC to find the root. 3. Use your graph or algebra to find the two asymptotes of the function. 2. Use your GDC to find the y-intercept.

Example (a) Write down the values of x for which g(x) = 0.

Example As g(x) is in the denominator of the function, a and b are the values of x for which g(x) = 0. These values were found in part (a). a = -0.790 and b = 1.79 as a < b.

Example continued…

Example continued… d The degree of g(x) is greater than the degree of the numerator. Therefore, when x gets very large, the term 2x^3/g(x) approaches zero and f(x) will approach the value of 1.

Domain and range The domain and range will be examined further in later lessons. For the purpose of this slide you may assume that: Domain is the x-values of a function or graph (input). Range is the y-values of a function or graph (output).

Domain and range Domain Range Domain Range 5 5 5 8 1 8 11 -5 7 -11 20 -3 -1 1 -3 This is a one-to-one relationship, and can be a function with an inverse. This is a many-to-one relationship, and is a function but without an inverse.

Domain and range continued 1 -1 -2 1 undefined -1 3 3 6 -3 -1 4 -3 13 -4 This is a one-to-many relationship, and therefore is not a function.

A one-to-one relationship is a function with an inverse. Domain and range continued A one-to-one relationship is a function with an inverse. Domain Range 1 A one-to-many relationship is a not a function. 2 3 -1 A many-to-one relationship is a function without an inverse. 10 5 Function - a rule that links each member of the domain to exactly one member of the range. This is a many-to-one relationship, and is a function but without an inverse.

Domain and range finder Cut out the graphs and the domain and range finder. For the domain, we want to look at the x coordinates. Use the left and right flaps. For the range, we want to look at the y coordinates. Use the top and bottom flaps.

Domain and range finder Use the flaps to help you see what domain and range the graph has.

Domain and range finder Function Domain Range

Restricting the domain However by stating the domain the function becomes one-to-one and has an inverse. This is a function without an inverse, as can be see from the graph.

Restricting the range This is not a function as it is a one-to-many relationship as can be seen from the graph. However, by restricting the range:

Domain & Range

Set vs. Interval Notation

Set vs. Interval Notation

Set vs. Interval Notation

Set vs. Interval Notation

You should know… The line x = a is a vertical asymptote if f(a) is undefined and Horizontal asymptotes tell us how the graph of a function behaves when x gets very large positively or negatively. They are often found in rational and exponential functions.

Be prepared… Know the major features of your GDC. When asked to sketch a graph, make effective use of the zoom and trace features to ensure that you are looking at an accurate representation of the graph. Set your window equal to the given domain When finding numerical values such as maximum and minimum points, use the appropriate menus and not the trace features of your GDC.

Information about sketching graphs You may be asked to sketch a graph. This is usually after you have used a GDC, but not always. A sketch does not need to be accurate, but it does need to be neat and it must show the vital information. You will not need graph paper. Your sketch should be: about a third of a page in size, drawn in pencil with a ruler for all the straight lines including the axes, clearly labeled, this includes the axes. On your sketch you must mark the following points: where the graph intercepts the axes, draw and label the asymptotes.

Sketching graphs Sketch the following graphs: about a third of a page in size, drawn in pencil with a ruler for all the straight lines including the axes, clearly labeled, this includes the axes. On your sketch you must mark the following points: where the graph intercepts the axes, draw and label the asymptotes.