Problem 58 - 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. $$ y=x^{2}-1, \quad \text { compressed horizontally by a factor of } 2 $$

Step-by-Step Solution

Answer

The equation is \( y = 4x^2 - 1 \) with a horizontal compression by a factor of 2.

Step 1: Identify the Transformation

We are given a function \( y = x^2 - 1 \) and the transformation described as a horizontal compression by a factor of 2. This means that for each x-value, the effect of the function is compressed towards the y-axis.

Step 2: Understand Horizontal Compression

A horizontal compression by a factor of 2 means that each x-coordinate of the graph is multiplied by 1/2. For a function \( f(x) \), the new function after horizontal compression is \( f(2x) \).

Step 3: Apply the Transformation

Substitute \(2x\) for \(x\) in the original function. This gives the new function: \[ y = (2x)^2 - 1 \].

Step 4: Simplify the Function

Calculate \((2x)^2\) which equals \(4x^2\). Thus, the transformed function becomes \[ y = 4x^2 - 1 \].

Key Concepts

Horizontal Compression

When we talk about horizontal compression, we're discussing how a graph shrinks towards the y-axis. Imagine squeezing the graph from the sides without changing its height. This happens when the x-values of a function are transformed.

• A horizontal compression by a factor of 2 means all x-values are halved.
• In mathematical terms, every x in your function is replaced with \( cx \), where \( c \) is the compression factor, less than 1. Here, \( c = \frac{1}{2} \).

To apply this to a function like \( y = x^2 - 1 \):

• Replace each \( x \) with \( \frac{x}{2} \) (or equivalently, multiply by 2) to compress it horizontally.
• For the function \( y = x^2 - 1 \), substitute to get \( y = (2x)^2 - 1 \).
• This means the new x-values will adjust the spread of the graph on the coordinate plane.

Horizontal compression doesn't affect the y-values directly, but it makes the parabola appear steeper as it becomes narrower.

Quadratic Function

A quadratic function is a polynomial function on the form \( y = ax^2 + bx + c \). These functions produce a characteristic U-shaped curve called a parabola.

• In the function \( y = x^2 - 1 \), there is no \( b \) term, suggesting the parabola is symmetric around the origin.
• The parabola's vertex is at \( (0, -1) \), meaning it's shifted one unit down from the classic \( y = x^2 \).

Quadratics are simpler to graph because:

• The 'a' value determines the direction and width of the parabola (if \( a > 0 \), it opens upwards; if \( a < 0 \), it opens downwards).
• The larger the absolute value of \( a \), the narrower the parabola.

This understanding helps interpret transformations like horizontal compressions by predicting how the graph's shape will alter.

Graph Transformation

Graph transformation involves changing the graph's position, size, or shape without altering its core properties. Transformations include dilations (compressions, stretches), translations (slides), reflections, and rotations.
In our example of \( y = x^2 - 1 \), the focus is on horizontal compression:

• The graph's width decreases, making the parabola appear steeper.
• Key points along the x-axis move towards the y-axis, conserving the overall shape.
• Practically, it's shown by transforming from \( x^2 \) to \( (2x)^2 \).

Other transformations could include vertical stretches (altering the y-values), translations (moving the graph up, down, left, or right), and reflections (flipping across axes).
Understanding these transformations is crucial for graphing accuracy and interpreting how changes holistically affect a function's overall representation.