Vertical Stretch

Definition

Vertical stretch is a transformation that changes the amplitude or scale of a function along the y-axis. It involves multiplying the function by a constant value, which can either expand or compress the function vertically, effectively changing its range and graphical appearance.

5 Must Know Facts For Your Next Test

1. Vertical stretch is a type of dilation that affects the function's range and amplitude, but not its domain.
2. The vertical stretch factor, denoted as 'a' in the function $f(x) = a \cdot g(x)$, determines the degree of expansion or compression along the y-axis.
3. If the vertical stretch factor 'a' is greater than 1, the function is expanded vertically, and if 'a' is between 0 and 1, the function is compressed vertically.
4. Vertical stretch can be used to model real-world phenomena, such as the relationship between force and displacement in Hooke's law or the growth of a population over time.
5. Understanding vertical stretch is crucial for analyzing and graphing various function families, such as linear, quadratic, exponential, and trigonometric functions.

Review Questions

• How does the vertical stretch factor 'a' affect the graph of a function?
  • The vertical stretch factor 'a' in the function $f(x) = a \cdot g(x)$ determines the degree of vertical expansion or compression of the graph. If 'a' is greater than 1, the function is stretched vertically, and if 'a' is between 0 and 1, the function is compressed vertically. The amplitude of the function, which represents the distance between the midline and the maximum or minimum value, is also affected by the vertical stretch factor.
• Explain how vertical stretch can be used to model real-world phenomena.
  • Vertical stretch is used to model various real-world relationships, such as the force-displacement relationship in Hooke's law, where the force is proportional to the displacement of a spring. In this case, the vertical stretch factor represents the spring constant, which determines the degree of vertical expansion or compression of the force-displacement graph. Additionally, vertical stretch can be used to model population growth over time, where the vertical stretch factor represents the growth rate of the population.
• Analyze the effects of vertical stretch on the graphical representation and characteristics of different function families.
  • Vertical stretch can significantly impact the graphical representation and characteristics of various function families. For linear functions, vertical stretch changes the slope and y-intercept, affecting the steepness and position of the line. In quadratic functions, vertical stretch alters the parabola's opening, height, and vertex. For exponential functions, vertical stretch adjusts the growth or decay rate, and for trigonometric functions, it modifies the amplitude of the sine, cosine, or tangent curves. Understanding how vertical stretch affects these function families is crucial for analyzing and graphing their behavior accurately.

Related terms

Amplitude:

The maximum absolute value of a function, which represents the distance between the midline and the maximum or minimum value of the function.

Transformation: A change in the form or appearance of a function, such as translation, reflection, dilation, or vertical/horizontal stretch.

Dilation: A transformation that changes the size of a function, either expanding or compressing it, without changing its overall shape.