Sure! I can help you prepare for a unit test on functions and their graphs. Below are some potential topics and questions that may be included in such a test:
Topics to Review
- Types of Functions: Linear, quadratic, polynomial, rational, exponential, logarithmic, trigonometric, etc.
- Function Notation: Understanding how to read and write functions using function notation \( f(x) \).
- Domain and Range: Finding the domain and range of various functions.
- Intercepts: Finding x-intercepts and y-intercepts.
- Transformations of Functions: Shifting, reflecting, stretching, and compressing graphs.
- Composition of Functions: Calculating and understanding the composition \( (f \circ g)(x) \).
- Inverse Functions: Finding the inverse of a function and understanding its graph.
- Graphing Functions: Plotting points and sketching their graphs accurately.
- Asymptotes and Behavior at Infinity: Finding horizontal and vertical asymptotes and understanding end behavior.
- Piecewise Functions: Evaluating and graphing piecewise-defined functions.
Sample Questions
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Identify the Function Type:
- Is \( f(x) = 2x^2 - 4x + 1 \) a linear, quadratic, polynomial, or another type of function?
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Find Domain and Range:
- Determine the domain and range of \( f(x) = \frac{1}{x-3} \).
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Intercepts:
- Find the x-intercept(s) and y-intercept of \( f(x) = x^2 - 5x + 6 \).
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Graph Transformation:
- Describe the transformations of \( g(x) = f(x - 3) + 2 \) where \( f(x) \) is the basic function \( f(x) = x^2 \).
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Composition of Functions:
- Given \( f(x) = 3x + 1 \) and \( g(x) = x^2 \), find \( (f \circ g)(x) \) and \( (g \circ f)(x) \).
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Find the Inverse:
- Find the inverse of the function \( f(x) = 2x - 5 \).
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Graphing a Function:
- Sketch the graph of \( f(x) = |x - 2| + 3 \).
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Behavior at Infinity:
- Determine the end behavior of the function \( f(x) = -2x^3 + x^2 + 4 \).
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Piecewise Function Evaluation:
- Evaluate the piecewise function ( f(x) = \begin{cases} x^2 & \text{if } x < 0 \ 2x + 1 & \text{if } x \geq 0 \end{cases} ) for \( x = -3 \) and \( x = 2 \).
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Asymptotes:
- Find the vertical and horizontal asymptotes of the function \( g(x) = \frac{2x + 1}{x^2 - 4} \).
Practice Problems
- Graph the function \( f(x) = \sqrt{x + 2} - 1 \) and identify key features such as the vertex and intercepts.
- Given the function \( f(x) = 3\sin(x) \), describe its amplitude, period, and graph one complete cycle.
Feel free to ask for explanations or practice on specific areas where you need the most assistance!