Problem 57 - 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. $$ y=x^{2}-1, \quad \text { stretched vertically by a factor of } 3 $$

Step-by-Step Solution

Answer

The graph is stretched vertically by a factor of 3, giving the equation \( y = 3x^2 - 3 \).

Step 1: Identify the Original Function

The original function given is \( y = x^2 - 1 \). This is a standard quadratic function with its vertex at \((-1, 0)\) shifted down by 1 unit.

Step 2: Determine Transformation Type

The function is to be stretched vertically by a factor of 3. This means every point on the graph will be multiplied by 3 in the vertical direction.

Step 3: Apply the Vertical Stretch

To apply the vertical stretch, multiply the entire function by 3. This transforms the original function \( y = x^2 - 1 \) into \( y = 3(x^2 - 1) \).

Step 4: Distribute the Stretch Factor

Distribute the stretch factor 3 through the equation: \( y = 3x^2 - 3 \). This is the new equation after the vertical stretch.

Key Concepts

Vertical Stretch

A vertical stretch is a transformation that affects the height of a graph. When we vertically stretch a function, we effectively "pull" the graph taller or shorter along the y-axis, depending on the factor provided. This transformation is achieved by multiplying the output (or y-value) of the entire function by a constant factor. The factor, which is greater than one in a vertical stretch, increases the distance each point on the graph from the x-axis. For the function \( y = x^2 - 1 \), applying a vertical stretch by a factor of 3 transforms the equation to \( y = 3(x^2 - 1) \).

• The vertex initially at \(-1\) vertically moves to \(-3\), making the graph steeper.
• The shape of the graph is amplified in the upward and downward directions.

Understanding the concept of vertical stretch is crucial, as it changes the perceived "influence" or impact of certain values on the graph, making some points more pronounced.

Quadratic Function

Quadratic functions play a vital role in mathematics, characterized by the form \( y = ax^2 + bx + c \), where \( a \), \( b \), and \( c \) are constants, and \( a eq 0 \). They produce parabolic graphs that can open either upwards or downwards, depending on the sign of \( a \). For our exercise, the quadratic function given is \( y = x^2 - 1 \). This particular quadratic has a simple form with its vertex at the origin, shifted down to \((0, -1)\).

• The "a" value, when positive, indicates an upward-opening parabola.
• Shifts upwards or downwards do not change the parabola's shape, merely its position.
• Key features include the vertex, axis of symmetry, and the direction of the graph.

Grasping the basics of quadratic functions helps you predict graph shapes and their transformations, such as stretches and shifts, which are pivotal in understanding advanced mathematical concepts.

Transformation Factor

The transformation factor is a crucial aspect of graph transformations, dictating how much the graph's shape and position change. In graph theory, a transformation factor can stretch, compress, shift, or reflect a graph—altering its appearance without changing its core characteristics. In the exercise, the transformation factor given is a vertical stretch by 3.

• When the factor is greater than 1, like 3, we stretch the graph.
• This transformation multiples each y-value by the factor.
• The larger the factor, the more pronounced the stretch is, making the function steeper.

Transformation factors are potent tools in mathematics, allowing us to experiment and understand how graphs behave under various modifications. Mastering their application is key in solving complex equations and understanding real-world phenomena through visual representation.