Problem 39 - Algebra and Trigonometry with Analytic Geometry

Question

Exer. 33-40: Explain how the graph of the function compares to the graph of \(y=f(x)\). For example, for the equation \(y=2 f(x+3)\), the graph of \(f\) is shifted 3 units to the left and stretched vertically by a factor of 2 . $$ y=-2 f\left(\frac{1}{3} x\right) $$

Step-by-Step Solution

Answer

The graph of \(f(x)\) is stretched horizontally by 3, reflected over the x-axis, and stretched vertically by 2.

Step 1: Identify Components

Analyze the given function \(y = -2f\left(\frac{1}{3}x\right)\) to identify the transformations. This equation indicates a multiplication by the factor 2, a negative sign, and a compression factor of \(\frac{1}{3}\).

Step 2: Horizontal Compression/Stretch

The term \(\frac{1}{3}x\) in the function indicates a horizontal stretch, because \(\frac{1}{3}\) is less than 1. Replace \(x\) with \(3x\) to find that the graph is stretched horizontally by a factor of 3.

Step 3: Vertical Reflection and Stretch

The factor \(-2\) in front of the \(f\) is responsible for a vertical stretch by a factor of 2 and a reflection across the x-axis since it's negative. This means the graph of \(f(x)\) is flipped upside down and stretched.

Key Concepts

Horizontal Stretch

When analyzing functions, it's important to understand the impact of transformations on their graphs. A horizontal stretch occurs when the variable inside the function's argument (usually the "x" value) is divided by a factor. This manipulation changes how wide the graph appears.

For example, in the exercise given, the function is described as \(y=-2f\left(\frac{1}{3}x\right)\). The term \(\frac{1}{3}x\) highlights a horizontal stretch. Here, the fraction \(\frac{1}{3}\) suggests replacing \(x\) with \(3x\), which stretches the graph horizontally by a factor of 3. This means the graph will appear wider. Essentially, points on the graph are moved further from the y-axis, leading to an expanded appearance.

To summarize:

• The fraction inside the function's argument causes the width change.
• To find the new width, take the reciprocal of the fraction.
• A fraction less than 1 inside leads to a wider graph appearance.

Vertical Reflection

A vertical reflection involves flipping the graph over a horizontal axis. In mathematical terms, this usually means changing the sign in front of the function. This changes the direction in which the graph opens.

In the function \(y=-2f\left(\frac{1}{3}x\right)\), the negative sign (\(-\)) before the "2" indicates a reflection across the x-axis. This is like looking at the graph in a mirror placed along the x-axis, reflecting all points below the axis to above, and vice-versa.

Key Points:

• A negative coefficient in front of the function indicates a vertical reflection.
• All y-values of the graph invert their sign.
• The shape of the graph remains unchanged, only its orientation is flipped.

Vertical Stretch

Vertical stretches change the height of the graph of a function, affecting how steep or flat the graph appears. Multiplying the entire function by a number stretches or compresses it vertically depending on the factor.

In our case, \(y=-2f\left(\frac{1}{3}x\right)\), the factor "2" applies to the entire function, meaning a vertical stretch by this factor occurs. This transformation makes the graph's peaks and valleys more pronounced, effectively pulling the graph away or toward the x-axis, depending on the factor's magnitude.

Important Considerations:

• A factor greater than 1 results in a vertical stretch.
• This stretch increases the amplitude of the graph.
• The steepness of the graph changes but not its x-span or horizontal position.