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A horizontal stretch happens when a function is stretched away from the y-axis. This means that we make the x-values take longer to reach their destination points on the graph. To achieve a horizontal stretch by a factor of \( c \), you replace every \( x \) in the function with \( \frac{x}{c} \). For instance, stretching the function \( y = 1 - x^3 \) horizontally by a factor of 2 involves the substitution of \( x \) with \( \frac{x}{2} \).
As a result, the new equation becomes \( y = 1 - \left(\frac{x}{2}\right)^3 \). This new equation represents the function stretched by the desired factor and gives us a wider graph along the x-axis than before. It's important to note that a factor greater than 1 will stretch the graph, whereas a factor between 0 and 1 will compress it.

A function graph is a visual representation of the relationship between variables in your function, usually \( x \) and \( y \). It helps us see how changes in variables affect the output, allowing for better understanding of the function's behavior. With a polynomial function like \( y = 1 - x^3 \), its graph shows us a smooth and continuous curve. Polynomial graphs can cross the x-axis and change direction, creating interesting shapes.
By transforming the graph—such as stretching it horizontally—we change how the function looks but not the fundamental relationships between the variables. This is a powerful tool to visualize how functions "act" when manipulated!
Using graphing software or plotting by hand, seeing transformations in action gives a deeper insight into mathematical functions.

A polynomial function is a mathematical expression made up of variables and coefficients, where the variables are raised to whole-number powers. An example is \( y = 1 - x^3 \), which is a cubic polynomial because its highest exponent is 3. Polynomial functions have various properties such as:

• They can have different shapes and sizes depending on their leading coefficient.
• They always form a continuous line or curve.
• They can have multiple roots, or points where the graph crosses the x-axis.

Polynomials are versatile tools in mathematics as they model many real-world situations and offer a base for more advanced mathematics. Understanding how polynomial graphs transform—using stretches, compressions, and other transformations—helps in interpreting the function's real-world implications.

When transforming equations, we change the appearance of the function's graph while keeping its basic properties intact. The transformation equation tells us exactly how a graph is altered. For a horizontal stretch, the transformation modifies the input variable by replacing \( x \) with \( \frac{x}{c} \). This makes the graph "wider" because it takes more x-value to produce the same y-value.
The transformation equation for a horizontally stretched version of \( y = 1 - x^3 \) is \( y = 1 - \frac{x^3}{8} \). Here, the "8" comes from substituting and simplifying \( \left(\frac{x}{2}\right)^3 \) which equals \( \frac{x^3}{8} \).
Understanding transformation equations is crucial because it provides a consistent method for graph manipulation, finding real-world function applications and improving analytical skills.