Vertical Shift

Definition

A vertical shift is a transformation that moves a function up or down on the coordinate plane, without changing the shape or orientation of the function. This concept is important in understanding how to graph and manipulate both quadratic and logarithmic functions.

5 Must Know Facts For Your Next Test

1. In a quadratic function of the form $f(x) = a(x - h)^2 + k$, the parameter 'k' represents the vertical shift, moving the parabola up or down on the coordinate plane.
2. For logarithmic functions of the form $f(x) = a + b\log_c(x)$, the parameter 'a' determines the vertical shift, translating the graph up or down.
3. Vertical shifts do not affect the overall shape or orientation of the function, only its position on the y-axis.
4. Positive vertical shift values move the function upward, while negative values move it downward on the coordinate plane.
5. Understanding vertical shifts is crucial for accurately graphing and analyzing the behavior of both quadratic and logarithmic functions.

Review Questions

• Explain how the parameter 'k' in the quadratic function $f(x) = a(x - h)^2 + k$ affects the graph of the function.
  • The parameter 'k' in the quadratic function $f(x) = a(x - h)^2 + k$ represents the vertical shift of the parabola. When 'k' is positive, the graph is shifted upward on the y-axis, and when 'k' is negative, the graph is shifted downward. The shape and orientation of the parabola remain unchanged, but its position on the coordinate plane is altered by the vertical shift.
• Describe the effect of the parameter 'a' on the graph of the logarithmic function $f(x) = a + b\log_c(x)$.
  • In the logarithmic function $f(x) = a + b\log_c(x)$, the parameter 'a' determines the vertical shift of the graph. When 'a' is positive, the graph is shifted upward on the y-axis, and when 'a' is negative, the graph is shifted downward. The shape of the logarithmic curve, which is determined by the base 'c' and the coefficient 'b', remains unchanged, but its position on the coordinate plane is affected by the vertical shift represented by the parameter 'a'.
• Analyze the similarities and differences between the effects of vertical shifts on quadratic and logarithmic functions.
  • Both quadratic and logarithmic functions can undergo vertical shifts, which move the graph up or down on the coordinate plane without changing the overall shape or orientation of the function. In quadratic functions, the vertical shift is represented by the parameter 'k' in the form $f(x) = a(x - h)^2 + k$, while in logarithmic functions, the vertical shift is determined by the parameter 'a' in the form $f(x) = a + b\log_c(x)$. The key similarity is that positive shift values move the graph upward, and negative values move it downward. The primary difference is that the vertical shift in quadratic functions affects the position of the parabola, while in logarithmic functions, it translates the entire curve up or down on the y-axis.

Related terms

Translation: A transformation that moves a function in the horizontal and/or vertical direction without changing its shape.

Transformation: A change made to the graph of a function that alters its appearance, such as shifting, stretching, or reflecting the function.

Parent Function: The original, untransformed version of a function, which serves as the basis for applying various transformations.