Absolute value functions are mathematical functions that measure the distance of a number from zero on a number line, regardless of direction. These functions are characterized by their V-shaped graphs, which reflect their unique behavior around the origin, resulting in distinct transformations when graphed. They exhibit properties such as symmetry and piecewise linearity, making them essential in understanding various graphing techniques and transformations.
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The standard form of an absolute value function is given by $$f(x) = a|x - h| + k$$, where (h, k) is the vertex of the graph.
The parameter 'a' affects the vertical stretch or compression and the direction of the opening of the graph; if 'a' is positive, the V opens upwards, and if negative, it opens downwards.
Absolute value functions are always non-negative; they cannot produce negative outputs because they measure distance.
The graph of an absolute value function will always be symmetric about the vertical line that passes through its vertex.
Transformations such as shifting left or right (using h) or up and down (using k) allow for modifications to the basic shape of the absolute value graph.
Review Questions
How does changing the parameter 'a' in the absolute value function affect its graph?
'a' in the function $$f(x) = a|x - h| + k$$ alters both the direction in which the graph opens and its steepness. If 'a' is greater than 1, the graph stretches vertically, making it steeper, while if '0 < a < 1', it compresses vertically, making it wider. Additionally, if 'a' is negative, it reflects the graph across the x-axis, changing its orientation completely.
Explain how to identify and determine the vertex of an absolute value function from its equation.
The vertex of an absolute value function written in standard form $$f(x) = a|x - h| + k$$ can be identified as the point (h, k). This point represents either the maximum or minimum value of the function depending on whether 'a' is positive or negative. By locating this vertex on a graph, one can establish the turning point where the V-shape of the function begins to change direction.
Evaluate how transformations impact an absolute value function's graph and provide an example.
Transformations significantly alter the appearance and position of an absolute value function's graph. For example, consider the function $$f(x) = 2|x - 3| + 1$$. Here, shifting right by 3 units and up by 1 unit places its vertex at (3, 1). The vertical stretch factor of 2 means that each y-value will be twice as high compared to its basic form. Such transformations allow for flexible modeling of various real-world situations using absolute value functions.
The point where the two linear pieces of an absolute value function meet, typically at the lowest or highest point of the graph.
Transformations: Changes made to the graph of a function, including translations, reflections, stretches, and compressions that affect the position and shape of the graph.
Piecewise Function: A function defined by multiple sub-functions, each applicable to specific intervals or conditions, which is how absolute value functions can be expressed mathematically.