Defining Absolute Maximum and Minimum Values
Absolute maximum and minimum values of a function — also called global extrema — are the highest and lowest y-values across the entire domain, a key topic in enVision Algebra 1 Chapter 10 for Grade 11. A function has a maximum value when some y-value is greater than or equal to all others; it has a minimum when some y-value is less than or equal to all others. For example, f(x) = -x² + 4 has an absolute maximum of 4 at x = 0 but no minimum; g(x) = |x| - 2 has an absolute minimum of -2 but no maximum; h(x) = sin(x) has both a maximum of 1 and a minimum of -1.
Key Concepts
Property A function has a maximum value when there is a $y$ value that is greater than or equal to all other $y$ values in the function''s range. A function has a minimum value when there is a $y$ value that is less than or equal to all other $y$ values in its range. These values are also known as global or absolute maximums and minimums.
Examples The function $f(x) = x^2 + 4$ has a maximum value of $4$. It has no minimum value. The function $g(x) = |x| 2$ has a minimum value of $ 2$. It has no maximum value. The function $h(x) = \sin(x)$ has a maximum value of $1$ and a minimum value of $ 1$.
Explanation The maximum and minimum are the "highest" and "lowest" points on the entire graph of a function. The maximum value is the largest output ($y$ value) the function can produce, while the minimum value is the smallest output. Not all functions have a maximum or minimum value; for example, a line like $y=x$ continues infinitely in both positive and negative $y$ directions.
Common Questions
What is an absolute maximum of a function?
An absolute maximum is the largest y-value the function ever reaches across its entire domain — a value that is greater than or equal to all other y-values in the range.
Does every function have an absolute maximum and minimum?
No. For example, y = x has no maximum or minimum because its outputs extend to positive and negative infinity. Functions like parabolas opening downward have a maximum but no minimum.
What is the absolute minimum of g(x) = |x| - 2?
The absolute minimum is -2, occurring at x = 0. Since |x| ≥ 0 for all x, the smallest possible output is 0 - 2 = -2.
What are the absolute maximum and minimum of h(x) = sin(x)?
The absolute maximum is 1 and the absolute minimum is -1. The sine function oscillates between these two values for all real x.
Why does f(x) = -x² + 4 have a maximum but no minimum?
The parabola opens downward (a = -1 < 0), reaching its peak at the vertex (0, 4). As x moves away from 0 in either direction, f(x) decreases without bound, so there is no minimum value.