Graphing Functions in Mathematics AA SL
Sketching Graphs from Given Information
Sketching graphs allows students to visualize functions and their properties. Unlike precise drawing, sketching involves creating a rough representation of a function's behavior.
TipWhen sketching, focus on key features such as intercepts, turning points, and asymptotes rather than plotting many individual points.
To sketch a graph effectively:
- Identify the function type (e.g., linear, quadratic, exponential)
- Determine key points (y-intercept, x-intercepts if any)
- Find any turning points or asymptotes
- Consider the function's behavior as x approaches positive and negative infinity
- Draw the axes and plot the key features
- Connect the points with a smooth curve that reflects the function's behavior
For the quadratic function $f(x) = x^2 - 4x + 3$:
- y-intercept: (0, 3)
- x-intercepts: solve $x^2 - 4x + 3 = 0$ to get x = 1 and x = 3
- Vertex: $(-b/(2a), f(-b/(2a))) = (2, -1)$
- Sketch:
Using Technology to Graph Functions
Graphing calculators and software can quickly plot functions, including sums and differences of functions for more complicated graphs.
NoteAlways verify technological outputs with your mathematical understanding to catch potential errors or misinterpretations.
Steps for using technology to graph functions:
- Enter the function into the graphing tool
- Set an appropriate viewing window
- Adjust the scale if necessary
- Use built-in features to find key points (zeros, extrema, etc.)
- Interpret the graph in the context of the problem
To graph $g(x) = \sin(x) + \cos(x)$ using a graphing calculator:
- Enter the function
- Set the window to show at least one full period, e.g., $x: [-2\pi, 2\pi]$, $y: [-2, 2]$
- Click "Graph"
Labeling Graphs
Proper labelling is crucial for clear mathematical communication. All graphs should include:
- Labeled $x$ and $y$ axes with appropriate scales
- Title or function equation
- Key features marked (e.g., intercepts, turning points, asymptotes)
Students often forget to label axes or omit units, which can lead to misinterpretation of the graph.
Drawing vs. Sketching
Understanding the difference between "draw" and "sketch" is important in exam contexts:
- Draw: Create a precise, accurate graph using appropriate tools (ruler, protractor)
- Sketch: Produce a freehand drawing showing key features without precise measurements
In exams, read instructions carefully to determine whether a sketch or a precise drawing is required.
Graphing Various Function Types
Students should be prepared to graph a wide range of functions, including those not explicitly covered elsewhere in the syllabus. This might include:
- Polynomial functions of higher degree
- Rational functions
- Piecewise functions
- Absolute value functions
- Logarithmic and exponential functions
- Trigonometric functions and their inverses
Sketching a piecewise function:
$$f(x) = \begin{cases} x^2 & \text{ if } x < 0 \\ 2x + 1 & \text{ if } x \geq 0 \end{cases}$$
- For $x < 0$, sketch a parabola
- For $x \geq 0$, sketch a line
- Ensure continuity at x = 0
Advanced Graphing Techniques
As students progress, they will develop skills in:
- Transformations of functions (translations, reflections, stretches)
- Combining functions (addition, subtraction, multiplication, division)
- Composition of functions
- Inverse functions and their graphs.
