Problem 63 - 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. $$ y=\sqrt{4-x^{2}}, \text { stretched horizontally by a factor of } 2 $$

Step-by-Step Solution

Answer

The stretched graph equation is \( y = \frac{1}{2}\sqrt{16 - x^2} \).

Step 1: Identify the Horizontal Stretch Factor

We are given the function \( y = \sqrt{4-x^2} \) and instructed to stretch it horizontally by a factor of 2. A horizontal stretch by a factor of 2 implies moving each point on the graph away from the y-axis, doubling the horizontal coordinate values. This occurs by replacing \( x \) with \( \frac{x}{2} \) in the equation.

Step 2: Transform the Function for Horizontal Stretch

To apply a horizontal stretch by a factor of 2 to the function \( y = \sqrt{4-x^2} \), we replace \( x \) with \( \frac{x}{2} \). This results in the new equation: \[ y = \sqrt{4 - \left(\frac{x}{2}\right)^2}. \] Simplifying within the square root gives \[ y = \sqrt{4 - \frac{x^2}{4}}. \]

Step 3: Simplify the Stretched Equation

The expression \( 4 - \frac{x^2}{4} \) can be rewritten by finding a common denominator: \( \frac{16}{4} - \frac{x^2}{4} = \frac{16 - x^2}{4} \). This simplifies our equation to \[ y = \sqrt{\frac{16 - x^2}{4}}. \] Further simplification within the square root leads to \[ y = \frac{1}{2}\sqrt{16 - x^2}. \]

Step 4: State the Final Stretched Function

The final equation of the graph after a horizontal stretch by a factor of 2 is obtained as \( y = \frac{1}{2}\sqrt{16 - x^2} \). This accounts for the described transformation.

Key Concepts

Horizontal Stretch

A horizontal stretch is a type of graph transformation that alters the horizontal component of a function's graph. To understand a horizontal stretch, imagine each point on the graph being pushed away from the y-axis. The distance each point is from the y-axis is increased by a certain factor, known as the stretch factor.

For example, in the function provided, the equation is given as \( y = \sqrt{4-x^2} \), and we're asked to stretch it horizontally by a factor of 2. This means every x-coordinate of the points on the graph is doubled. In general, when stretching horizontally by a factor \( a \), we replace \( x \) with \( \frac{x}{a} \).

• This replacement results in the graph becoming wider, or stretched along the horizontal axis.
• The x-values change, but the y-values remain the same, purely affecting the graph's horizontal aspect.

Understanding horizontal stretches helps us manipulate functions to match desired shapes or real-world data more closely.

Function Transformation

Function transformation involves changing the appearance or position of a graph without altering its inherent shape. This includes operations such as translating, reflecting, stretching, and compressing. For functions, transformations allow us to adjust variables or scale graphs in useful ways.

In our example, applying a horizontal stretch to the original function involves adjusting the variable within the square root. By replacing \( x \) with \( \frac{x}{2} \), the function is effectively stretched horizontally. This replacement exemplifies how transformations modify functions:

• By substituting variables, we adjust the graph's scale or position.
• The transformed function, \( y = \frac{1}{2}\sqrt{16 - x^2} \), achieves the desired stretch.

Overall, understanding transformations allows us to predict and control how graphs will change with different modifications.

Square Root Function

The square root function is one of the core components of many mathematical transformations. It's important to understand this function to fully grasp the concept of graph transformations.

The basic form of a square root function is \( y = \sqrt{x} \), which produces a graph that starts at the origin and curves upwards to the right. In the problem exercise, the square root function is slightly more complex: \( y = \sqrt{4-x^2} \).

• This function gives a half-circle shape, limited by the domain \(-2 \leq x \leq 2\).
• The function value represents the upper half of a circle with radius 2, centered at the origin.

Understanding how the square root function behaves allows us to predict changes when transformations such as stretches or compressions are applied. Being familiar with its properties is crucial for working with more complex transformations.