The maximum or minimum of an absolute value function depends on the function's form and any transformations applied to it. Specifically, the vertex of the absolute value function represents either the minimum or maximum value.
Understanding Absolute Value Functions
An absolute value function is generally expressed as f(x) = a|x - h| + k, where:
- a determines the direction of opening and any vertical stretch or compression.
- (h, k) represents the vertex of the function.
Minimum Value
- If a > 0, the absolute value function opens upwards, and the vertex (h, k) represents the minimum value of the function. The minimum value is k.
Maximum Value
- If a < 0, the absolute value function opens downwards, and the vertex (h, k) represents the maximum value of the function. The maximum value is k.
No Overall Maximum or Minimum
It's important to note that absolute value functions that open upwards (a > 0) extend infinitely upwards, and therefore, there's no absolute maximum. Similarly, absolute value functions that open downwards (a < 0) extend infinitely downwards and thus have no absolute minimum. Instead, when people speak of a maximum value of a downwards facing absolute value function, they are referring to the highest point, which occurs at the vertex. The "minimum" of an upwards-facing absolute value function refers to the lowest point at the vertex.
Examples
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f(x) = |x - 2| + 3: Here, a = 1 (positive), so the function opens upwards. The vertex is (2, 3), representing a minimum value of 3.
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f(x) = -2|x + 1| - 4: Here, a = -2 (negative), so the function opens downwards. The vertex is (-1, -4), representing a maximum value of -4.
Finding the Vertex
To find the maximum or minimum, you need to determine the vertex (h, k). This can be done by:
- Rewriting the absolute value function in the form f(x) = a|x - h| + k.
- Or, if the function is in a different form, by completing the square or using other algebraic techniques to transform it into the standard form.
In conclusion, the absolute value function has either a minimum or maximum value located at its vertex. The sign of a determines whether the vertex represents a minimum (a > 0) or a maximum (a < 0). Keep in mind that upwards-facing functions do not technically have an overall maximum, and downwards-facing functions do not technically have an overall minimum.
