Problem 63 - A Graphical Approach to College Algebra

Question

Concept Check If \((r, 0)\) is an \(x\) -intercept of the graph of \(y=f(x),\) what statement can be made about an \(x\) -intercept of the graph of each function? (Hint: Make a sketch.) (a) \(y=-f(x)\) (b) \(y=f(-x)\) (c) \(y=-f(-x)\)

Step-by-Step Solution

Answer

\((r, 0)\) is an x-intercept for all three functions.

Step 1: Understand the X-Intercept

If \((r, 0)\) is an \(x\)-intercept of the graph of \(y=f(x)\), it means that when \(x = r\), \(y = 0\). Therefore, \(f(r) = 0\). This forms the basis of our analysis for each transformation of the function.

Step 2: Analyze Part (a): y = -f(x)

The function \(y = -f(x)\) represents a vertical reflection of \(y = f(x)\) across the x-axis. Since reflection across the x-axis does not affect x-intercepts, the x-intercept remains the same. Thus, \((r, 0)\) is also an x-intercept of \(y = -f(x)\).

Step 3: Analyze Part (b): y = f(-x)

The function \(y = f(-x)\) represents a reflection across the y-axis. However, since x-intercepts are points where \(y = 0\), they are not affected by horizontal reflection directly. Therefore, \((r, 0)\) remains an x-intercept for \(y = f(-x)\).

Step 4: Analyze Part (c): y = -f(-x)

The function \(y = -f(-x)\) is a combined transformation: a reflection across the y-axis followed by a reflection across the x-axis. Despite these transformations, the nature of x-intercepts means they remain unaffected, as \(y\) remains 0 at these points. Therefore, \((r, 0)\) is an x-intercept of \(y = -f(-x)\) as well.

Key Concepts

Function Transformation

Function transformation means changing the appearance or position of a graph. These transformations can include translations, reflections, stretches, and compressions. Understanding how these changes affect the original function helps us predict new graph positions or shapes.

Common transformations include:

• Translations: Shifting the graph horizontally or vertically.
• Reflections: Flipping the graph across axes.
• Stretches/Compressions: Expanding or compressing the graph vertically or horizontally.

When dealing with x-intercepts, transformations such as reflections can have specific effects, which we'll discuss further in this article.

Reflection Across Axes

A reflection across axes involves flipping a function's graph over a specific axis, either the x-axis or the y-axis. This transformation can change the position of points on the graph.

When reflecting across an axis:

• Across the x-axis: The y-values of each point become negative, effectively turning the graph upside down.
• Across the y-axis: The x-values of each point are mirrored, swapping the graph's left and right sides.

These transformations are crucial when analyzing how functions behave under different conditions, particularly when observing how they affect x-intercepts.

Vertical Reflection

Vertical reflection flips a graph over the x-axis. This is represented by the function \( y = -f(x) \). When applying a vertical reflection:

• The y-values of all points on the graph become their negative counterparts.
• Visually, the graph is turned upside down.

Despite this flip, the x-intercepts of the function remain unchanged. This is because vertical reflection alters only the y-values. Since x-intercepts occur where y=0, no change occurs at these points. Thus, in the transformation from \( y = f(x) \) to \( y = -f(x) \), the x-intercepts stay the same.

Horizontal Reflection

Horizontal reflection involves flipping the graph across the y-axis, represented by the change from \( y = f(x) \) to \( y = f(-x) \). Here's what happens:

• The x-values are mirrored over the y-axis, transforming every point \( (x, y) \) into \( (-x, y) \).
• This results in a left-right flip of the entire graph.

Even though horizontal reflection alters the x-values at which y-values are achieved, it doesn’t affect the x-intercepts directly. This is because x-intercepts are the x-values where y equals zero, and flipping over the y-axis doesn't change the fact that y remains zero at these intercepts. Hence, x-intercepts remain \( (r, 0) \) even after horizontal reflection.