Graph each function. See Examples 1 and 2. ƒ(x)=3|x|
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Identify the basic function: The function is a transformation of the absolute value function .
Understand the transformation: The coefficient 3 in vertically stretches the graph of by a factor of 3.
Determine the vertex: The vertex of the absolute value function is at the origin (0,0). Since there are no horizontal or vertical shifts, the vertex of is also at (0,0).
Plot key points: Choose a few values for (e.g., -2, -1, 0, 1, 2) and calculate for each to get points like (-2,6), (-1,3), (0,0), (1,3), (2,6).
Draw the graph: Plot the points on a coordinate plane and connect them with straight lines to form a 'V' shape, ensuring the graph is symmetric about the y-axis.
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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|, measures the distance of a number from zero on the number line, always yielding a non-negative result. For any real number x, |x| equals x if x is positive or zero, and -x if x is negative. This function creates a V-shaped graph that opens upwards, with its vertex at the origin (0,0). Understanding this function is crucial for graphing ƒ(x)=3|x|.
A vertical stretch occurs when a function is multiplied by a factor greater than one, affecting the steepness of its graph. 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 alters the y-values of the function, resulting in a graph that rises more quickly as x moves away from zero.
Graphing techniques involve plotting points and understanding transformations to accurately represent a function's behavior visually. For ƒ(x)=3|x|, one can start by plotting key points such as (0,0), (1,3), and (-1,3), then apply the vertical stretch to these points. Familiarity with these techniques allows for a more efficient and accurate graphing process, essential for visualizing the function's characteristics.