Problem 106 - A Graphical Approach to Precalculus with Limits

Question

If we are given the graph of \(y=f(x)\), we can obtain the graph of \(y=-f(x)\) by reflecting across the \(x\)-axis, and we can obtain the graph of \(y=f(-x)\) by reflecting across the y-axis.You are given the graph of a rational function \(y=f(x)\). Draw \(a\) sketch by hand of the graph of (a) \(y=-f(x)\) and (b) \(y=f(-x)\) (Check your book to see graph)

Step-by-Step Solution

Answer

Reflect the graph over the x-axis for \(y=-f(x)\) and over the y-axis for \(y=f(-x)\).

Step 1: Understanding Reflection across x-axis

Reflecting the graph of \(y=f(x)\) across the x-axis to get \(y=-f(x)\) involves flipping the graph upside down. For every point \((x, y)\) on the original graph, the new graph will have a point \((x, -y)\). This means that all y-values of the function \(y=f(x)\) become negative or have their sign changed in \(y=-f(x)\). Begin by sketching the original graph and then draw its reflection with all the y-values inverted.

Step 2: Sketching Reflection across x-axis

After understanding how to apply the reflection rule, sketch the graph. Locate key points on the graph of \(f(x)\), such as the intercepts and any given points, and invert their y-coordinates. Draw the new graph based on these inverted points ensuring it maintains the same horizontal positioning on the x-axis.

Step 3: Understanding Reflection across y-axis

Reflecting the graph of \(y=f(x)\) across the y-axis to get \(y=f(-x)\) requires changing the direction of the x-values. For every point \((x, y)\) on the original graph, the new graph will have a point \((-x, y)\). This operation flips the graph horizontally without altering the y-values. Take each x-coordinate and switch its sign to create a mirror image of the graph across the y-axis.

Step 4: Sketching Reflection across y-axis

Utilize the insights from the reflection across the y-axis to sketch the graph of \(y=f(-x)\). Identify significant points from the original graph and reverse their x-coordinates. Ensure all structural aspects of the graph (like asymptotes or turning points) are accounted for, and draw the new graph reflecting vertically over the y-axis.

Key Concepts

Reflection Across X-Axis

When reflecting a graph across the x-axis, the visual effect is similar to flipping the graph over the x-axis. In essence, all points that were above the x-axis are moved below, and those below are moved above.
To achieve this mathematically, you take the function \(y = f(x)\) and convert it into \(y = -f(x)\). Every y-coordinate on the original graph changes sign. For example, a point \((x, y)\) on the original graph becomes \((x, -y)\) on the transformed graph.

• This means if a point was at (3, 4), it now moves to (3, -4).
• If the graph crosses the x-axis, the intercept remains unchanged, as the point (x, 0) stays the same.

Through this transformation, the graph's shape remains unchanged, but its orientation is inverted. This reflection works particularly well with rational functions since they often have asymptotic lines that follow the axis. Identifying those and inverting the relevant points can help sketch an accurate transformed graph.

Reflection Across Y-Axis

Reflecting across the y-axis is a horizontal mirroring transformation. Here, the function \(y = f(x)\) is turned into \(y = f(-x)\), meaning we change the x-coordinates to their opposites. Visually, any point on the right of the y-axis moves to the left, and vice versa.
This transformation adjusts the graph's symmetry over the y-axis.

• For a point \((x, y)\) on the original graph, the new point will be \((-x, y)\).
• If a point was (5, 2) on \(y = f(x)\), it transforms to (-5, 2) on \(y = f(-x)\).
• This alteration retains the y-values while flipping the direction of x-values.

If the graph has any symmetry about the y-axis before this transformation, that property will remain after. This is often seen in even functions where \(f(x) = f(-x)\), though it extends to more complex graphs like rational functions as well.

Rational Function

Rational functions are quotients of two polynomials, expressed as \(y = \frac{P(x)}{Q(x)}\), where \(P(x)\) and \(Q(x)\) are polynomials and \(Q(x) eq 0\). They are characterized by their unique behavior due to the existence of vertical and horizontal asymptotes.
Vertical asymptotes occur where the denominator \(Q(x)\) approaches zero, causing the function’s value to approach infinity. Horizontal asymptotes are found by examining the leading coefficients of the polynomials as they define the function’s behavior as \(x\) gets very large or very small.

• Often, rational functions will have points or regions where the function is undefined.
• They may intersect the axes at points where either \(P(x) = 0\) or \(Q(x) = 0\).

Rational functions can demonstrate a wide range of transformations when manipulated graphically. Understanding the base function’s graph helps predict its behavior when subjected to reflections across axes or other transformations.