key term - Vertical Transformation

Definition

A vertical transformation is a type of transformation applied to a function that results in a shift of the function's graph in the vertical direction. This transformation can either stretch, compress, or reflect the graph of the function along the y-axis, depending on the specific parameters of the transformation.

5 Must Know Facts For Your Next Test

1. The vertical transformation of a function is represented by the form $f(x) = a \cdot g(x) + k$, where $a$ is the vertical stretch or compression factor, and $k$ is the vertical shift.
2. If $a > 1$, the graph of the function is stretched vertically. If $0 < a < 1$, the graph is compressed vertically.
3. If $a < 0$, the graph of the function is reflected across the x-axis.
4. The vertical shift, $k$, determines how much the graph is shifted up or down on the y-axis.
5. Vertical transformations are particularly important in the study of quadratic functions, as they allow for the creation of a wide variety of parabolic shapes.

Review Questions

• Explain how the parameters $a$ and $k$ in the equation $f(x) = a \cdot g(x) + k$ affect the vertical transformation of a function's graph.
  • The parameter $a$ determines the vertical stretch or compression of the function's graph. If $a > 1$, the graph is stretched vertically, and if $0 < a < 1$, the graph is compressed vertically. If $a < 0$, the graph is reflected across the x-axis. The parameter $k$ determines the vertical shift of the graph, moving it up or down on the y-axis. Together, these two parameters allow for a wide range of vertical transformations to be applied to the original function's graph.
• Describe how vertical transformations can be used to create different shapes of parabolic graphs for quadratic functions.
  • Vertical transformations are particularly important in the study of quadratic functions, as they allow for the creation of a wide variety of parabolic shapes. By adjusting the parameters $a$ and $k$ in the equation $f(x) = a \cdot g(x) + k$, the graph of a quadratic function can be stretched, compressed, or reflected vertically, resulting in parabolas that are tall, short, or even flipped upside down. This flexibility in the shape of the parabola is crucial for modeling a diverse range of real-world phenomena that can be represented by quadratic functions.
• Analyze how the vertical transformation of a function's graph can affect the interpretation and application of the function in a given context.
  • The vertical transformation of a function's graph can significantly impact the interpretation and application of the function in a given context. For example, in the case of a quadratic function representing the height of an object over time, a vertical stretch could indicate a higher maximum height, while a vertical compression could suggest a lower maximum height. Similarly, a vertical reflection could change the direction of the motion, such as an object moving downward instead of upward. Understanding how the parameters $a$ and $k$ affect the vertical transformation is essential for accurately interpreting the behavior and characteristics of the function within the specific context it is being applied to.

Related terms

Dilation: A dilation is a transformation that stretches or compresses a function's graph along the y-axis, without affecting the x-axis.

Reflection:

A reflection is a transformation that flips the graph of a function across the x-axis or y-axis, resulting in a mirror image.

Quadratic Function:

A quadratic function is a polynomial function of degree two, with the general form $f(x) = ax^2 + bx + c$, where $a$, $b$, and $c$ are real numbers.