Table of contents
- 0. Review of College Algebra4h 43m
- 1. Measuring Angles39m
- 2. Trigonometric Functions on Right Triangles2h 5m
- 3. Unit Circle1h 19m
- 4. Graphing Trigonometric Functions1h 19m
- 5. Inverse Trigonometric Functions and Basic Trigonometric Equations1h 41m
- 6. Trigonometric Identities and More Equations2h 34m
- 7. Non-Right Triangles1h 38m
- 8. Vectors2h 25m
- 9. Polar Equations2h 5m
- 10. Parametric Equations1h 6m
- 11. Graphing Complex Numbers1h 7m
4. Graphing Trigonometric Functions
Graphs of the Sine and Cosine Functions
Problem 15c
Textbook Question
Graph each function. See Examples 1 and 2. ƒ(x) = 3|x|
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Step 1: Recognize that the function is a transformation of the basic absolute value function .
Step 2: Understand that the coefficient 3 in front of indicates a vertical stretch of the graph by a factor of 3. This means that for every point on the graph of , the corresponding point on will be .
Step 3: Identify the vertex of the graph. Since there are no horizontal or vertical shifts, the vertex remains at the origin .
Step 4: Plot key points. For example, calculate for to get points .
Step 5: Draw the graph by connecting the points with straight lines, forming a 'V' shape that opens upwards, with the vertex at the origin and the arms of the 'V' being steeper than those of due to the vertical stretch.
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Key Concepts
Here are the essential concepts you must grasp in order to answer the question correctly.
Absolute Value Function
The absolute value function, denoted as |x|, outputs the non-negative value of x regardless of its sign. This means that for any real number x, |x| is equal to x if x is positive or zero, and -x if x is negative. Understanding this function is crucial for graphing, as it creates a V-shaped graph that opens upwards, reflecting the symmetry about the y-axis.
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Vertical Stretch
A vertical stretch occurs when a function is multiplied by a constant factor greater than one. In the function ƒ(x) = 3|x|, the factor of 3 stretches the graph vertically by a factor of 3, making it steeper than the basic absolute value function. This transformation affects the y-values of the function, increasing them proportionally while maintaining the overall shape of the graph.
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Graphing Techniques
Graphing techniques involve plotting points and understanding transformations to visualize functions accurately. For ƒ(x) = 3|x|, one can start by plotting key points such as (0,0), (1,3), and (-1,3), then connect these points to form the V-shape. Familiarity with transformations, such as shifts and stretches, is essential for accurately representing the function on a coordinate plane.
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Related Practice
Textbook Question
In Exercises 1–6, determine the amplitude of each function. Then graph the function and y = sin x in the same rectangular coordinate system for 0 ≤ x ≤ 2π.
y = 4 sin x
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