key term - F(x) = a|x - h| + k

Definition

The function f(x) = a|x - h| + k is an absolute value function, where 'a' represents the vertical stretch or shrink, 'h' represents the horizontal shift, and 'k' represents the vertical shift. This function can be used to model a variety of real-world situations involving absolute values.

5 Must Know Facts For Your Next Test

1. The absolute value function f(x) = a|x - h| + k has a V-shaped graph, with the vertex located at the point (h, k).
2. The graph of the function f(x) = a|x - h| + k is symmetric about the vertical line x = h.
3. The domain of the function f(x) = a|x - h| + k is all real numbers, and the range is [k - |a|, k + |a|].
4. The function f(x) = a|x - h| + k is continuous for all real values of x.
5. The function f(x) = a|x - h| + k is increasing on the interval (-∞, h) and decreasing on the interval (h, ∞).

Review Questions

• Explain how the parameter 'a' in the function f(x) = a|x - h| + k affects the graph of the function.
  • The parameter 'a' in the function f(x) = a|x - h| + k determines the vertical stretch or shrink of the graph. If 'a' is greater than 1, the graph is vertically stretched by a factor of 'a'. If 'a' is between 0 and 1, the graph is vertically shrunk by a factor of 'a'. The absolute value of 'a' also determines the range of the function, as the range is [k - |a|, k + |a|].
• Describe the effect of the parameter 'h' in the function f(x) = a|x - h| + k on the graph of the function.
  • The parameter 'h' in the function f(x) = a|x - h| + k determines the horizontal shift of the graph. If 'h' is positive, the graph is shifted to the right by 'h' units. If 'h' is negative, the graph is shifted to the left by 'h' units. The vertex of the V-shaped graph is located at the point (h, k), and the graph is symmetric about the vertical line x = h.
• Analyze the behavior of the function f(x) = a|x - h| + k as the input variable 'x' approaches positive and negative infinity.
  • As the input variable 'x' approaches positive infinity, the function f(x) = a|x - h| + k approaches positive infinity, regardless of the values of 'a', 'h', and 'k'. This is because the absolute value function |x - h| approaches positive infinity as 'x' gets larger. Conversely, as 'x' approaches negative infinity, the function f(x) = a|x - h| + k approaches negative infinity, again regardless of the parameter values. This behavior is a result of the absolute value function's properties, where the distance from zero on the number line increases without bound as the input variable moves away from the horizontal shift 'h'.

Related terms

Absolute Value:

The absolute value of a number is its distance from zero on the number line, and is denoted by the symbol '|x|'.

Vertical Stretch/Shrink: The parameter 'a' in the function f(x) = a|x - h| + k determines the vertical stretch or shrink of the graph, where 'a' > 1 causes a vertical stretch and 0 < a < 1 causes a vertical shrink.

Horizontal Shift:

The parameter 'h' in the function f(x) = a|x - h| + k determines the horizontal shift of the graph, where a positive value of 'h' shifts the graph to the right and a negative value of 'h' shifts the graph to the left.