Problem 3 - Algebra and Trigonometry

Question

Fill in the blank with the appropriate axis (x-axis or \(y\) -axis) (a) The graph of \(y=-f(x)\) is obtained from the graph of \(y=f(x)\) by reflecting in the_________ (b) The graph of \(y=f(-x)\) is obtained from the graph of \(y=f(x)\) by reflecting in the_________

Step-by-Step Solution

Answer

(a) x-axis; (b) y-axis.

Step 1: Understanding Reflection

To solve these problems, we need to understand how reflections work on the coordinate plane. A reflection over the x-axis means flipping the graph over the x-axis, while a reflection over the y-axis means flipping it over the y-axis.

Step 2: Analyzing Part (a)

For the expression \(y = -f(x)\), the transformation involves multiplying the output or \(y\)-value by \(-1\). This changes the sign of the output and thus flips the graph over the x-axis.

Step 3: Analyzing Part (b)

For the expression \(y = f(-x)\), the transformation involves replacing \(x\) with \(-x\). This changes the sign of the input or \(x\)-values, which results in flipping the graph over the y-axis.

Key Concepts

Reflection Over The X-Axis

Reflections are a type of transformation that alter the position of a graph in the coordinate plane. A reflection over the x-axis is specifically about switching the graph upside down.
When you reflect a function over the x-axis, each point on the graph is mirrored across the x-axis.
This means that for a function represented as \( y = f(x) \), the reflected function becomes \( y = -f(x) \). When you apply this transformation:

• The y-coordinates of each point change sign.
• If the original point is \( (x, y) \), then the reflected point is \( (x, -y) \).

This kind of reflection is like flipping a picture upside down over a line running horizontally across the page.
A helpful tip to remember: when you see a negative sign in front of the function, it indicates a reflection over the x-axis.

Reflection Over The Y-Axis

While reflection over the x-axis impacts the output or y-values of a function, reflection over the y-axis affects the input, or x-values.
Think of it as flipping the graph horizontally instead of vertically.
In mathematical terms, if you start with \( y = f(x) \), reflecting over the y-axis will transform it to \( y = f(-x) \). This type of reflection acts as follows:

• The x-coordinates of each point change sign.
• If your initial point is \( (x, y) \), the new reflected point will be \( (-x, y) \).

Imagine a mirror placed along the y-axis, reflecting every point on the graph to the opposite side horizontally.
A quick clue to recognize this transformation: if "\(-x\)" appears inside the function, expect a reflection over the y-axis.

Coordinate Plane

The Coordinate Plane is a critical concept in understanding graph transformations like reflections.
It's the workspace where we plot and analyze functions and their movements.
The plane is made up of two perpendicular axes: the x-axis (horizontal) and the y-axis (vertical).

• The x-axis stretches left and right.
• The y-axis goes up and down.

Each point on this plane has coordinates in a pair, written as \( (x, y) \). The origin, or the point where both axes intersect, is \( (0, 0) \).
Understanding the layout of the coordinate plane helps make sense of how reflections and other transformations alter the appearance and position of a graph.
When tackling math problems, visualize how the graph shifts or flips over these axes, aiding comprehension of reflections.