Absolute Value Function Shifted: \(y = |x – a|\)
The function \(y = |x – a|\) is a shift of the absolute value function along the x-axis. If \(a\) is positive, the graph shifts to the right by \(a\) units. If \(a\) is negative, the graph shifts to the left by \(a\) units.
Domain and Range
The domain of the function is all real numbers. This means you can put any number (including negatives) into the function. In interval notation, this is \((-∞, ∞)\). In set-builder notation, this is \(\{x | x ∈ ℝ\}\).
The range of the function is all numbers that are zero or more. This is because the absolute value of a number is always zero or more. In interval notation, this is \([0, ∞)\). In set-builder notation, this is \(\{y | y ∈ ℝ, y ≥ 0\}\).
Graph
The graph of the function \(y = |x – a|\) with \(a = 5\) as an example. The red line represents the domain of the function, and the blue line represents the range of the function.
Absolute Value Function Shifted: \(y = |x| + a\)
The function \(y = |x| + a\) is a shift of the absolute value function along the y-axis. If \(a\) is positive, the graph shifts up by \(a\) units. If \(a\) is negative, the graph shifts down by \(a\) units.
Domain and Range
The domain of the function is all real numbers. This means you can put any number (including negatives) into the function. In interval notation, this is \((-∞, ∞)\). In set-builder notation, this is \(\{x | x ∈ ℝ\}\).
The range of the function is all numbers that are \(a\) or more. This is because the absolute value of a number is always zero or more, and then we add \(a\). In interval notation, this is \([a, ∞)\). In set-builder notation, this is \(\{y | y ∈ ℝ, y ≥ a\}\).
Graph
The graph of the function \(y = |x| + a\) with \(a = 5\) as an example. The red line represents the domain of the function, and the blue line represents the range of the function.
Absolute Value Function Shifted: \(y = |x – a| + b\)
The function \(y = |x – a| + b\) is a shift of the absolute value function along both the x-axis and y-axis. If \(a\) is positive, the graph shifts to the right by \(a\) units. If \(a\) is negative, the graph shifts to the left by \(a\) units. If \(b\) is positive, the graph shifts up by \(b\) units. If \(b\) is negative, the graph shifts down by \(b\) units.
Domain and Range
The domain of the function is all real numbers. This means you can put any number (including negatives) into the function. In interval notation, this is \((-∞, ∞)\). In set-builder notation, this is \(\{x | x ∈ ℝ\}\).
The range of the function is all numbers that are \(b\) or more. This is because the absolute value of a number is always zero or more, and then we add \(b\). In interval notation, this is \([b, ∞)\). In set-builder notation, this is \(\{y | y ∈ ℝ, y ≥ b\}\).
Graph
The graph of the function \(y = |x – a| + b\) with \(a = 5\) and \(b = 3\) as an example. The red line represents the domain of the function, and the blue line represents the range of the function.
Absolute Value Function Transformed: \(y = a|x – b| + c\)
The function \(y = a|x – b| + c\) is a transformation of the absolute value function. The value of \(a\) affects the steepness of the graph, \(b\) shifts the graph along the x-axis, and \(c\) shifts the graph along the y-axis. If \(a\) is positive, the graph opens upwards, and if \(a\) is negative, the graph opens downwards. If \(b\) is positive, the graph shifts to the right by \(b\) units, and if \(b\) is negative, the graph shifts to the left by \(b\) units. If \(c\) is positive, the graph shifts up by \(c\) units, and if \(c\) is negative, the graph shifts down by \(c\) units.
Domain and Range
The domain of the function is all real numbers. This means you can put any number (including negatives) into the function. In interval notation, this is \((-∞, ∞)\). In set-builder notation, this is \(\{x | x ∈ ℝ\}\).
The range depends on the value of \(a\). If \(a\) is positive, the range is all numbers that are \(c\) or more. If \(a\) is negative, the range is all numbers that are \(c\) or less. In interval notation, this is \([c, ∞)\) for \(a > 0\) and \((-∞, c]\) for \(a < 0\). In set-builder notation, this is \(\{y | y ∈ ℝ, y ≥ c\}\) for \(a > 0\) and \(\{y | y ∈ ℝ, y ≤ c\}\) for \(a < 0\).
Graph
The graph of the function \(y = a|x – b| + c\) with \(a = 2\), \(b = 5\), and \(c = 3\) as an example. The red line represents the domain of the function, and the blue line represents the range of the function.
Sign Function: \(y = \frac{|x|}{x}\)
The function \(y = \frac{|x|}{x}\) is often referred to as the “sign function” because it essentially extracts the sign of the input \(x\). It returns -1 if \(x\) is negative, indicating that \(x\) is less than zero, and it returns 1 if \(x\) is positive, indicating that \(x\) is greater than zero. The function is undefined at \(x = 0\) because zero is neither positive nor negative.
Domain and Range
The domain of the function is all real numbers except zero. In interval notation, this is \((-∞, 0) ∪ (0, ∞)\). In set-builder notation, this is \(\{x | x ∈ ℝ, x ≠ 0\}\).
The range of the function is the set containing only the numbers -1 and 1. This means that the function can only output these two values. In set notation, this is \(\{-1, 1\}\).
Graph
The graph of the function \(y = \frac{|x|}{x}\). The red line represents the domain of the function, and the blue lines represent the range of the function.