Horizontal Shift

Definition

A horizontal shift is a transformation of a function that moves the graph of the function left or right along the x-axis, without changing the shape or orientation of the graph. This concept is applicable to various types of functions, including transformations of functions, absolute value functions, exponential functions, trigonometric functions, and the parabola.

5 Must Know Facts For Your Next Test

1. A horizontal shift of a function $f(x)$ to the left by $h$ units is represented by the function $f(x-h)$, while a shift to the right by $h$ units is represented by $f(x+h)$.
2. Horizontal shifts can affect the domain and range of a function, as well as the location of any asymptotes or critical points.
3. In the context of absolute value functions, a horizontal shift can change the location of the vertex, which is the point where the graph changes direction.
4. Horizontal shifts of exponential functions can affect the rate of growth or decay, as well as the location of the y-intercept.
5. For trigonometric functions, a horizontal shift can change the phase of the function, which determines the position of the graph relative to the origin.

Review Questions

• Explain how a horizontal shift affects the graph of a function.
  • A horizontal shift of a function $f(x)$ to the left by $h$ units is represented by the function $f(x-h)$, while a shift to the right by $h$ units is represented by $f(x+h)$. This transformation moves the graph of the function along the x-axis, without changing the shape or orientation of the graph. Horizontal shifts can affect the domain and range of a function, as well as the location of any asymptotes or critical points, depending on the type of function.
• Describe how a horizontal shift impacts the graph of an absolute value function.
  • For an absolute value function $f(x) = |x|$, a horizontal shift to the left by $h$ units is represented by $f(x-h) = |x-h|$, while a shift to the right by $h$ units is represented by $f(x+h) = |x+h|$. This horizontal shift changes the location of the vertex, which is the point where the graph changes direction. The shape of the graph, however, remains the same, with the arms of the absolute value function still forming a V-shape.
• Analyze the effect of a horizontal shift on the graph of an exponential function.
  • For an exponential function $f(x) = a^x$, a horizontal shift to the left by $h$ units is represented by $f(x-h) = a^{x-h}$, while a shift to the right by $h$ units is represented by $f(x+h) = a^{x+h}$. This horizontal shift can affect the rate of growth or decay of the function, as well as the location of the y-intercept. The shape of the exponential curve, however, remains the same, with the graph still exhibiting its characteristic exponential growth or decay pattern.

Related terms

Function Transformation: The process of modifying the graph of a function by applying various transformations, such as horizontal shifts, vertical shifts, reflections, and dilations.

Asymptote: A line that a graph of a function approaches but never touches, which can be affected by a horizontal shift.

Period: The distance between two consecutive points on a periodic function's graph, which can be changed by a horizontal shift.