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IBMaths: AA SLDPRevision Notes2. Functions2.3 Functions Toolkit2.3.3 Graphing Functions

Graphing Functions (DP IB Maths: AA SL)

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Dan

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Dan

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Graphing Functions

How do I graph the function y = f(x)?

  • A point space left parenthesis a comma space b right parenthesis lies on the graph space y equals f left parenthesis x right parenthesis if space f left parenthesis a right parenthesis equals b
  • The horizontal axis is used for the domain
  • The vertical axis is used for the range
  • You will be able to graph some functions by hand
  • For some functions you will need to use your GDC
  • You might be asked to graph the sum or difference of two functions
    • Use your GDC to graph space y equals f left parenthesis x right parenthesis plus g left parenthesis x right parenthesis or space y equals f left parenthesis x right parenthesis minus g left parenthesis x right parenthesis
    • Just type the functions into the graphing mode

What is the difference between “draw” and “sketch”?

  • If asked to sketch you should:
    • Show the general shape
    • Label any key points such as the intersections with the axes
    • Label the axes
  • If asked to draw you should:
    • Use a pencil and ruler
    • Draw to scale
    • Plot any points accurately
    • Join points with a straight line or smooth curve
    • Label any key points such as the intersections with the axes
    • Label the axes

How can my GDC help me sketch/draw a graph?

  • You use your GDC to plot the graph
    • Check the scales on the graph to make sure you see the full shape
  • Use your GDC to find any key points
  • Use your GDC to check specific points to help you plot the graph

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Key Features of Graphs

What are the key features of graphs?

  • You should be familiar with the following key features and know how to use your GDC to find them
  • Local minimums/maximums
    • These are points where the graph has a minimum/maximum for a small region
    • They are also called turning points
      • This is where the graph changes its direction between upwards and downwards directions
    • A graph can have multiple local minimums/maximums
    • A local minimum/maximum is not necessarily the minimum/maximum of the whole graph
      • This would be called the global minimum/maximum
    • For quadratic graphs the minimum/maximum is called the vertex
  • Intercepts
    • y­­ – intercepts are where the graph crosses the y-axis
      • At these points x = 0
    • x – intercepts are where the graph crosses the x-axis
      • At these points y = 0
      • These points are also called the zeros of the function or roots of the equation
  • Symmetry
    • Some graphs have lines of symmetry
      • A quadratic will have a vertical line of symmetry
  • Asymptotes
    • These are lines which the graph will get closer to but not cross
    • These can be horizontal or vertical
      • Exponential graphs have horizontal asymptotes
      • Graphs of variables which vary inversely can have vertical and horizontal asymptotes

Sketching Polynomials Notes Diagram 1

Examiner Tip

  • Most GDC makes/models will not plot/show asymptotes just from inputting a function
    • Add the asymptotes as additional graphs for your GDC to plot
    • You can then check the equations of your asymptotes visually
    • You may have to zoom in or change the viewing window options to confirm an asymptote
  • Even if using your GDC to plot graphs and solve problems sketching them as part of your working is good exam technique
    • Label the key features of the graph and anything else relevant to the question on your sketch

Worked example

Two functions are defined by

space f open parentheses x close parentheses equals x squared minus 4 x minus 5 and space g open parentheses x close parentheses equals 2 plus fraction numerator 1 over denominator x plus 1 end fraction.

a)
Draw the graph space y equals f left parenthesis x right parenthesis.

2-2-2-ib-ai-key-features-of-graphs-a-we-solution

b)
Sketch the graph space y equals g left parenthesis x right parenthesis.

2-2-2-ib-ai-key-features-of-graphs-b-we-solution

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Intersecting Graphs

How do I find where two graphs intersect?

  • Plot both graphs on your GDC
  • Use the intersect function to find the intersections
  • Check if there is more than one point of intersection

Solving Equations Graphically Notes Diagram 1

How can I use graphs to solve equations?

  • One method to solve equations is to use graphs
  • To solve space f left parenthesis x right parenthesis equals a
    • Plot the two graphs space y equals f left parenthesis x right parenthesis and space y equals a on your GDC
    • Find the points of intersections
    • The x-coordinates are the solutions of the equation
  • To solve space f left parenthesis x right parenthesis equals g left parenthesis x right parenthesis
    • Plot the two graphs space y equals f left parenthesis x right parenthesis and space space y equals g left parenthesis x right parenthesis on your GDC
    • Find the points of intersections
    • The x-coordinates are the solutions of the equation
  • Using graphs makes it easier to see how many solutions an equation will have

Examiner Tip

  • You can use graphs to solve equations
    • Questions will not necessarily ask for a drawing/sketch or make reference to graphs
    • Use your GDC to plot the equations and find the intersections between the graphs

Worked example

Two functions are defined by

space f left parenthesis x right parenthesis equals x cubed minus x and space g left parenthesis x right parenthesis equals 4 over x.

a)
Sketch the graph space y equals f left parenthesis x right parenthesis.

2-2-2-ib-ai-sl-intersecting-graphs-a-we-solution

b)
Write down the number of real solutions to the equation space x cubed minus x equals 2.

2-2-2-ib-ai-sl-intersecting-graphs-b-we-solution

c)
Find the coordinates of the points where space y equals f left parenthesis x right parenthesis and space y equals g left parenthesis x right parenthesis intersect.

2-2-2-ib-ai-sl-intersecting-graphs-c-we-solution

d)
Write down the solutions to the equation space x cubed minus x equals 4 over x.

2-2-2-ib-ai-sl-intersecting-graphs-d-we-solution

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    Dan

    Author: Dan

    Expertise: Maths

    Dan graduated from the University of Oxford with a First class degree in mathematics. As well as teaching maths for over 8 years, Dan has marked a range of exams for Edexcel, tutored students and taught A Level Accounting. Dan has a keen interest in statistics and probability and their real-life applications.