Horizontal stretch

Definition

A horizontal stretch is a transformation of a graph that alters its width by expanding it away from the y-axis. This transformation involves compressing the input values of a function by a factor, which results in the graph appearing wider. In the context of polynomial functions, horizontal stretches can significantly change the shape and intersections of the graph with the x-axis, impacting how the function behaves across its domain.

5 Must Know Facts For Your Next Test

1. A horizontal stretch occurs when the function is transformed using a factor of `k`, resulting in the new function form `f(kx)` where `k > 1` indicates a stretch.
2. The width of the graph increases as the input values are stretched, leading to changes in the distance between roots and intersections with the x-axis.
3. If `k` is less than 1, this results in a horizontal compression instead of a stretch, making the graph appear narrower.
4. Horizontal stretches do not affect the y-coordinates of points on the graph; only their x-coordinates are influenced.
5. Understanding horizontal stretches is crucial for sketching polynomial functions accurately, as it helps predict changes in key features like turning points and intercepts.

Review Questions

• How does a horizontal stretch affect the roots of a polynomial function?
  • A horizontal stretch affects the roots of a polynomial function by altering their distances from each other. When you apply a horizontal stretch using a factor greater than 1 (e.g., transforming `f(x)` into `f(kx)` where `k > 1`), it makes the graph wider. This means that the x-coordinates of the roots will increase in magnitude, potentially changing their positions along the x-axis while maintaining their respective y-values at zero.
• In what ways do horizontal stretches differ from vertical stretches in transforming polynomial functions?
  • Horizontal stretches differ from vertical stretches primarily in how they affect the graph's dimensions. A vertical stretch modifies the output values (y-coordinates) by multiplying them by a factor greater than 1, making the graph taller without changing its width. In contrast, a horizontal stretch modifies input values (x-coordinates), expanding or compressing the graph horizontally. While both transformations influence the overall shape and features of polynomial functions, they target different axes and impact various attributes such as root locations and turning points.
• Evaluate how understanding horizontal stretches can improve your ability to graph polynomial functions accurately.
  • Understanding horizontal stretches allows you to predict and visualize how polynomial functions change in shape and position when transformed. By knowing that applying a stretch will widen the graph and alter root distances without affecting y-values, you can better anticipate where the function intersects with axes and identify turning points. This insight enhances your ability to sketch graphs accurately and recognize key features quickly, leading to better problem-solving strategies when working with polynomial functions.

Related terms

Vertical Stretch:

A vertical stretch is a transformation that expands the graph of a function away from the x-axis by multiplying the output values by a factor greater than 1.

Transformation: A transformation refers to any operation that alters the position, size, or shape of a graph, including translations, stretches, and reflections.

Polynomial Function: A polynomial function is a mathematical expression involving a sum of powers in one or more variables multiplied by coefficients, characterized by its degree and leading coefficient.