Problem 66 - Thomas Calculus

Question

Exercises \(57-66\) tell by what factor and direction the graphs of the given functions are to be stretched or compressed. Give an equation for the stretched or compressed graph. $$ y=1-x^{3}, \quad \text { stretched horizontally by a factor of } 2 $$

Step-by-Step Solution

Answer

The graph is transformed to \( y = 1 - \frac{x^3}{8} \).

Step 1: Understand Horizontal Stretch

A horizontal stretch affects the input of a function. When a function is stretched horizontally by a factor of \( c \), each \( x \) value in the function is replaced by \( \frac{x}{c} \). Thus, if \( f(x) = y \), the transformed function becomes \( f\left(\frac{x}{c}\right) \).

Step 2: Apply the Horizontal Stretch to the Function

Given the function \( y = 1 - x^3 \), we apply the horizontal stretch by replacing \( x \) with \( \frac{x}{2} \). This gives us the new function:\[y = 1 - \left(\frac{x}{2}\right)^3.\]

Step 3: Simplify the Transformed Function

Next, simplify the expression \( \left(\frac{x}{2}\right)^3 \) which is equal to \( \frac{x^3}{8} \). Therefore, the equation for the stretched graph becomes:\[y = 1 - \frac{x^3}{8}.\]

Key Concepts

Function Transformation

Function transformation is a powerful way to manipulate graphs and analyze their characteristics. It applies changes like stretching, shrinking, shifting, or reflecting a function's graph. Function transformations are generally performed by altering either the function's input (horizontal transformations) or its output (vertical transformations).

When discussing horizontal stretches or compressions, you alter the input part of the function. This means you replace each instance of the variable with a fraction. For example, if you have a horizontal stretch by a factor of 2, you replace every 'x' with 'x/2'.

Function transformation helps us to understand how changes to the equation affect the graph of the function. This insight is crucial in analyzing mathematical models, predicting patterns, and solving real-life application problems effectively.

Cubic Functions

Cubic functions are polynomial functions of degree three, generally written in the form: \[ f(x) = ax^3 + bx^2 + cx + d \] where 'a', 'b', 'c', and 'd' are constants, and 'a' is not zero. Characteristically, cubic functions can have up to three real roots and can display interesting behaviors like turning points. These functions have an "S"-shaped curve, giving them a distinctive appearance.

Cubic functions can experience function transformations like stretches, shifts, and reflections. By applying these transformations, we alter the graph's shape, ensuring that new scenarios or conditions are modeled accurately. For example, in this exercise, the cubic equation is transformed through a horizontal stretch, resulting in the function:\[ y = 1 - \frac{x^3}{8} \] which modifies its graph appearance and characteristics. Understanding cubic functions and their transformations are essential for advanced algebraic problem-solving and graphing.

Graphing Transformations

Graphing transformations involve changing a graph's shape or position on a coordinate system. This can include translating, stretching, compressing, or reflecting the graph.

Horizontal transformations affect the x-coordinates of a function. When graphing a horizontal stretch, each x-coordinate is divided by the stretch factor. For example, stretching the function \( y = 1 - x^3 \) horizontally by a factor of 2 transforms it into \( y = 1 - \frac{x^3}{8} \). This stretch makes the graph appear wider, altering the function's appearance but not its fundamental nature.

Understanding graphing transformations helps in visualizing algebraic expressions and interpreting physical, social, or technological models. It enables students to predict graph shapes and behaviors after transformations, fostering a deeper comprehension of mathematical concepts.