key term - Horizontal Shift

Definition

A horizontal shift refers to the lateral movement of a graph or function along the x-axis, either to the left or to the right. This transformation affects the position of the graph without changing its shape or orientation.

5 Must Know Facts For Your Next Test

1. A horizontal shift to the left is represented by adding a positive value to the input variable (x), while a horizontal shift to the right is represented by subtracting a positive value from the input variable (x).
2. Horizontal shifts can affect the domain and range of a function, as the new graph may now include or exclude certain values on the x-axis.
3. Horizontal shifts are commonly used in the graphing of quadratic functions (topic 9.7) and logarithmic functions (topic 10.3) to model real-world scenarios and transform the shape of the graph.
4. The amount of the horizontal shift is often represented by a constant value added to or subtracted from the input variable (x) in the function equation.
5. Horizontal shifts can be combined with other transformations, such as vertical shifts, reflections, and dilations, to create more complex transformations of a graph or function.

Review Questions

• Explain how a horizontal shift affects the graph of a quadratic function.
  • When graphing quadratic functions using transformations (topic 9.7), a horizontal shift moves the entire parabolic graph left or right along the x-axis, without changing its shape or orientation. A horizontal shift to the left is represented by adding a positive value to the input variable (x), while a shift to the right is represented by subtracting a positive value from the input variable (x). This transformation can alter the domain and range of the function, as the new graph may now include or exclude certain values on the x-axis.
• Describe the effect of a horizontal shift on the graph of a logarithmic function.
  • When evaluating and graphing logarithmic functions (topic 10.3), a horizontal shift moves the entire graph left or right along the x-axis, without changing its shape or orientation. Similar to quadratic functions, a horizontal shift to the left is represented by adding a positive value to the input variable (x), while a shift to the right is represented by subtracting a positive value from the input variable (x). This transformation can also affect the domain and range of the logarithmic function, as the new graph may now include or exclude certain values on the x-axis.
• Analyze how a horizontal shift can be combined with other transformations to create more complex graphs of quadratic and logarithmic functions.
  • Horizontal shifts can be combined with other transformations, such as vertical shifts, reflections, and dilations, to create more complex transformations of quadratic and logarithmic functions. For example, in topic 9.7, a quadratic function could undergo a horizontal shift to the left or right, followed by a vertical shift up or down, and then a reflection about the x-axis. Similarly, in topic 10.3, a logarithmic function could be horizontally shifted, then vertically shifted, and finally dilated to create a more intricate graph. These combined transformations allow for greater flexibility in modeling real-world scenarios and shaping the graphs of these functions to fit the desired characteristics.

Related terms

Vertical Shift:

A vertical shift refers to the upward or downward movement of a graph or function along the y-axis, without changing its shape or orientation.

Transformation: A transformation is a change in the position, size, or shape of a graph or function, which can include horizontal shifts, vertical shifts, reflections, and dilations.

Parent Function:

The parent function is the original, untransformed version of a function, which serves as a reference point for understanding how transformations affect the graph.