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:
