Problem 62 - College Algebra

Question

For the following exercises, describe how the graph of each function is a transformation of the graph of the original function \(f\). $$ g(x)=-f(3 x) $$

Step-by-Step Solution

Answer

The graph of \( g(x) \) is \( f(x) \) compressed horizontally by \( \frac{1}{3} \) and reflected over the x-axis.

Step 1: Identify the Base Graph

The base function is given as \( f(x) \). The exercise asks about the transformation of this graph. Start by recognizing this original graph, \( y = f(x) \), as the basis for comparison.

Step 2: Horizontal Transformation

In the given function \( g(x) = -f(3x) \), focus first on \( f(3x) \). The term \( 3x \) suggests a horizontal transformation. Specifically, \( 3x \) implies a horizontal compression by a factor of \( \frac{1}{3} \), since the input \( x \) is multiplied by 3.

Step 3: Vertical Reflection

The negative sign in \( -f(3x) \) indicates a vertical reflection. This sign flips the graph over the x-axis. Thus, all the y-values of \( f(3x) \) are multiplied by \(-1\), resulting in the reflection.

Step 4: Combined Transformations

Combine the transformations. The graph of \( g(x) = -f(3x) \) will be the graph of \( f(x) \) first compressed horizontally by a factor of \( \frac{1}{3} \) and then reflected over the x-axis due to the negative sign.

Key Concepts

Horizontal Compression

When thinking about horizontal compressions, imagine squeezing the graph of a function towards the y-axis. This type of transformation changes how "spread out" the graph is horizontally. In our exercise, we look at the function \[ g(x) = -f(3x) \] and focus on the term \( f(3x) \). Here, the x-value is multiplied by 3, which is crucial for horizontal transformations.

To understand this better, remember that a horizontal compression occurs when an input in the function is multiplied by a constant greater than one. This compression factor is calculated as the reciprocal of this constant. In this case, because we're dealing with a multiplier of 3, the actual compression factor is \( \frac{1}{3} \).

• This means the entire graph "shrinks" towards the y-axis.
• Each point on the graph moves closer to the y-axis by a factor of \( \frac{1}{3} \).
• The x-coordinates become reduced; for instance, any point originally at \( x=1 \) will now appear at \( x=\frac{1}{3} \).

A practical way to demonstrate might be using software or manually plotting points, but keeping the idea that every horizontal distance is now a third of what it used to be.

Vertical Reflection

Vertical reflections are transformations that flip the graph of a function over the x-axis. In the function given by\[ g(x) = -f(3x) \] the negative sign outside the function \(-f(3x)\) dictates this transformation. The graph reflects vertically because every y-value of the original function now becomes its own opposite.

Let’s break down what this means:

• If a point on the original function is \((x, y)\), after reflection it becomes \((x, -y)\).
• Imagine a point that originally had a bad day on -5 (y-value). Now after the reflection, it's having a plus 5 day!
• The x-axis serves as the mirror line over which all points are flipped.

In practice, if you imagine the original function, every upward hill becomes a downward valley and vice versa.
So, if a curve was smiling upwards like a positive parabola, after reflecting it looks like a sad face frown.

Original Function

The original function, denoted by \( y = f(x) \), acts as our baseline or reference point for understanding the transformations applied to create \( g(x) \).

Prior to squishing or flipping the graph, it's important to visualize or graph \( y = f(x) \) to know what you're starting with. This graph provides a canvas for implementing transformations such as compressions or reflections.

When addressing a transformation exercise:

• Identify this basic shape—it might be linear, quadratic, or any other form.
• This basis helps realize what the altered function \( g(x) = -f(3x) \) should look like.
• An understanding of the basic form gives clarity as to how drastic or subtle the transformations will impact the graph shape and direction.

This original framework serves as a control against which we measure these transformative steps, allowing us to see the horizontal and vertical changes more clearly.