Graphing Functions (DP IB Maths: AI HL)

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20 April 2023

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A point $(a, b)$ lies on the graph $y = f (x)$ if $f (a) = b$
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 $y = f (x) + g (x)$ or $y = f (x) - g (x)$
- Just type the functions into the graphing mode

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

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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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

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

Two functions are defined by

$f (x) = x^{2} - 4 x - 5$ and $g (x) = 2 + \frac{1}{x + 1}$ .

Draw the graph

y = f (x)

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

Sketch the graph

y = g (x)

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

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Solving Equations Graphically Notes Diagram 1

One method to solve equations is to use graphs
To solve $f (x) = a$
- Plot the two graphs $y = f (x)$ and $y = a$ on your GDC
- Find the points of intersections
- The x-coordinates are the solutions of the equation
To solve $f (x) = g (x)$
- Plot the two graphs $y = f (x)$ and $y = g (x)$ 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

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