Problem 40 - BIG IDEAS MATH Algebra 2: Common Core Student Edition 2015

Question

\(f(x)=4 \csc x\); vertical stretch by a factor of 2 and a reflection in the \(x\)-axis

Step-by-Step Solution

Answer

The final transformed function after a vertical stretch by a factor of 2 and a reflection in the \(x\)-axis is \(h(x) = -8 \csc x\).

Step 1: Apply the vertical stretch

Applying a vertical stretch by a factor of 2 to the function \(f(x)=4 \csc x\) would mean multiplying the function by 2. This gives us a transformed function \(g(x) = 2 \cdot f(x) = 2 \cdot 4 \csc x = 8 \csc x\).

Step 2: Reflect in the \(x\)-axis

A reflection in the \(x\)-axis means flipping the function upside-down, which is achieved by multiplying by -1. This gives us the final transformed function \(h(x) = -1 \cdot g(x) = - 1 \cdot 8 \csc x = -8 \csc x\).

Key Concepts

Understanding Vertical Stretch

A vertical stretch is a transformation that affects the y-values of a function. It changes how tall or short a graph appears without altering its shape. To apply a vertical stretch by a factor, you multiply the function by that factor.
For example, if you have a function like \(f(x) = \csc x\), and you want to apply a vertical stretch by a factor of 2, which means making the function 'taller' by twice as much, you would multiply \(f(x)\) by 2.

• This gives us \(g(x) = 2 \cdot f(x) = 2 \csc x\).
• The function stretches vertically by scaling all y-values by 2.

In the original exercise, the function \(f(x) = 4 \csc x\) was already vertically stretched by 4. Applying another vertical stretch by a factor of 2 results in \(8 \csc x\), amplifying the effect, making the result taller than the original.

Exploring Reflection

Reflection is a type of transformation that flips the function across a specified axis. In this case, we are interested in reflecting a graph across the x-axis. Flipping a function vertically inverts its y-values, changing positive outputs to negative and vice versa.
For any function \(f(x)\), reflecting in the x-axis is done by multiplying the function by -1. This means the graph mirrors itself across the horizontal axis.

• After our graph from the vertical stretch, \(g(x) = 8 \csc x\), reflecting it gives \(-8 \csc x\).
• This operation turns all peaks into valleys and valleys into peaks along the x-axis.

In the context of trigonometric functions like the cosecant, this transformation affects how maximum and minimum values of the graph appear. It's important to visualize how these reflect transformations interact, especially since cosecant functions have asymptotes and undefined points.

The Cosecant Function Demystified

The cosecant function, denoted as \(\csc x\), is the reciprocal of the sine function and is defined as \(\csc x = \frac{1}{\sin x}\). Since it's a reciprocal of sine, \(\csc x\) is undefined wherever \(\sin x = 0\), leading to the presence of vertical asymptotes at these points in its graph (e.g., multiples of \(\pi\)).

• The graph of \(\csc x\) appears as a series of disjointed U-shaped (upside-down and right-side-up) curves.
• As the angle approaches a sine zero, the cosecant tends towards infinity or negative infinity, jumping to another branch across an asymptote.

When transforming a \(\csc x\) function through vertical stretches and reflections, each curve maintains its U-shape, but the height and orientation of each curve change according to transformations. Thus, understanding how these transformations affect critical points, like maximum values, minimum values, and asymptotes, helps in grasping the cosecant function's altered graph.