Problem 62 - 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. $$ y=\sqrt{x+1}, \quad \text { stretched vertically by a factor of } 3 $$

Step-by-Step Solution

Answer

The new equation is \( y = 3\sqrt{x+1} \).

Step 1: Understanding Vertical Stretch

To stretch a graph vertically means to multiply the output (or dependent variable, often 'y') by a factor. In this case, the factor given is 3.

Step 2: Applying Vertical Stretch to the Function

To apply the vertical stretch by a factor of 3, multiply the entire function by 3. Therefore, the function becomes: \[ y = 3 \cdot \sqrt{x+1} \] This results in the output values being tripled, creating a steeper slope for the graph while preserving its shape.

Step 3: Writing the New Equation

After applying the vertical stretch, the new equation of the function becomes:\[ y = 3 \cdot \sqrt{x+1} \]This equation represents the stretched graph.

Key Concepts

Function Transformation

Function transformation is a powerful tool in mathematics, allowing you to change the position and shape of graphs. When transforming functions, you can perform actions like translations, reflections, stretches, and compressions on a graph. These transformations can be applied to any basic function by modifying its equation. To transform a function, we typically manipulate either the 'x' variable or the output (often 'y'). In our example, we focused on a vertical transformation.

• Translation moves the graph without altering its shape.
• Reflection flips the graph over either the x-axis or y-axis.
• Stretching or compressing changes the slope or width of the graph.

Understanding these transformations is essential because they allow you to control the graph's appearance, making functions more versatile.

Graph Compression

Graph compression and stretching are specific transformations that change the visual sizing of a graph. Compression is when the graph narrows, while stretching is when it widens.
When dealing with graph compression or stretching, consider whether the manipulation is vertical or horizontal:

• **Vertical Stretch/Compression:** This involves multiplying the whole function by a constant. If the constant is greater than 1, it results in a vertical stretch, making the graph steeper. If it's between 0 and 1, it results in vertical compression.
• **Horizontal Stretch/Compression:** Involves changing the 'x' variable. Mutliplying 'x' by a factor greater than 1 compresses the graph horizontally, while a factor between 0 and 1 stretches it horizontally.

In our case, we applied a vertical stretch by a factor of 3, which made the graph of the function steeper but preserved its overall shape. This is because each function output (or 'y' value) is now three times larger than it was for the same 'x' values.

Square Root Function

The square root function is a fundamental mathematical function denoted as \( y = \sqrt{x} \). It is a special case of the power function, where the exponent is 1/2.

• The basic shape of the square root function graph is a gentle curve that starts at the origin (0,0) and rises upwards.
• This curve represents the positive square root of 'x' and is defined only for non-negative values of 'x'.
• The graph continually increases, but at a decreasing rate, meaning it gets less steep as 'x' increases.

In the specific example \( y = \sqrt{x+1} \), adding 1 inside the square root shifts the graph 1 unit to the left. This transformation allows us to explore variations of the fundamental square root graph while still maintaining its inherent properties.