Problem 51 - Thomas Calculus

Question

Exercises \(51-60\) 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 . $$ y=x^{2}-1, \text { stretched vertically by a factor of } 3 $$

Step-by-Step Solution

Answer

The stretched graph is given by the equation \( y = 3x^2 - 3 \).

Step 1: Understand the Problem

We have the function \( y = x^2 - 1 \) and we need to vertically stretch this graph by a factor of 3. A vertical stretch by a factor of \( a \) means replacing \( y \) with \( ay \).

Step 2: Apply Vertical Stretch

To stretch the graph vertically, multiply the entire function \( y = x^2 - 1 \) by 3. This changes the function to \( y = 3(x^2 - 1) \).

Step 3: Simplify the New Equation

Simplify the stretched function: distribute 3 to each term inside the parentheses. The equation becomes \( y = 3x^2 - 3 \).

Step 4: Confirm Changes

The graph of \( y = x^2 - 1 \) is stretched vertically, so that every point \((x, y)\) is transformed to \((x, 3y)\). Thus, the new equation is \( y = 3x^2 - 3 \).

Key Concepts

Vertical Stretch

A vertical stretch is a type of graph transformation that affects the shape of a function's graph. When a function undergoes a vertical stretch, every point on the graph is moved away from or closer to the x-axis, depending on the factor by which it is stretched.

To perform a vertical stretch:

• Identify the function you are working with, such as the standard quadratic function, which is often in the form of \( y = ax^2 + bx + c \).
• Determine the stretch factor, which tells you how much to multiply the entire function by.
• Replace \( y \) with \( ay \), where \( a \) is the vertical stretch factor.

For example, if we have the function \( y = x^2 - 1 \) and we want to apply a vertical stretch by a factor of 3, we multiply each term in the function by 3, resulting in the new function: \( y = 3x^2 - 3 \). In this transformation, all original y-values are scaled by 3, making the graph taller.

Quadratic Functions

Quadratic functions are fundamental in algebra and appear in the general form \( y = ax^2 + bx + c \). These functions characterize parabolas, which are symmetric curves that can open either upwards or downwards, depending on the sign of the leading coefficient \( a \).

Key characteristics of quadratic functions include:

• The vertex, which is the highest or lowest point on the graph, located at \( x = -\frac{b}{2a} \).
• The axis of symmetry, a vertical line that passes through the vertex and splits the parabola into two mirror images.
• The directionality, determined by the sign of \( a \). If \( a > 0 \), the parabola opens upwards; if \( a < 0 \), it opens downwards.

In our example, the original quadratic function is \( y = x^2 - 1 \), which describes a parabola opening upwards with its vertex at the point \( (0, -1) \). When applying a vertical stretch, these properties remain but the graph gets taller or shorter, accentuating the vertical changes.

Graph Transformations

Graph transformations allow us to modify the appearance of a function's graph without changing its basic form. These transformations can include stretches, compressions, translations, and reflections.

Transformations help visualize changes by applying mathematical operations to the function’s equation:

• Vertical stretches and compressions (like multiplying \( y \) by a factor of \( a \)) change how wide or narrow the graph appears.
• Horizontal stretches and compressions (involving changes to the \( x \) term) adjust the width of the graph along the x-axis.
• Translations shift the graph up, down, left, or right.
• Reflections flip the graph over the x-axis or y-axis.

In the original problem, the transformation applied was a vertical stretch. This altered the graph’s height relative to the x-axis but maintained the parabola's symmetrical properties and orientation. Understanding these transformations can assist in graphing functions more accurately and predicting how changes will affect their graphs.