Problem 7 - Precalculus Mathematics for Calculus

Question

Suppose the graph of \(f\) is given. Describe how the graph of each function can be obtained from the graph of \(f\). (a) \(y=-f(x)\) (b) \(y=f(-x)\)

Step-by-Step Solution

Answer

(a) Reflect over x-axis. (b) Reflect over y-axis.

Step 1: Understanding y = -f(x)

To transform the graph of a function to obtain the graph of \( y = -f(x) \), you need to reflect the original graph over the x-axis. This is because changing the sign of the entire function negates all the output values, flipping the graph upside down.

Step 2: Reflecting over the x-axis

Take each point on the original graph, find its y-coordinate, and change its sign. If a point \((x, y)\) is on the graph of \( f(x) \), the corresponding point \((x, -y)\) will be on the graph of \( y = -f(x) \). Perform this transformation for all points on the graph.

Step 3: Understanding y = f(-x)

For the transformation \( y = f(-x) \), reflect the original graph over the y-axis. This transformation affects the input values (x-coordinates) by negating them, which mirrors the graph horizontally across the y-axis.

Step 4: Reflecting over the y-axis

To perform this reflection, take each point on the original graph, find its x-coordinate, and change its sign. A point \((x, y)\) on the graph of \( f(x) \) becomes \((-x, y)\) on the graph of \( y = f(-x) \). Apply this to all points on the original function graph.

Key Concepts

Reflection over the x-axis

When we talk about reflecting a graph over the x-axis, we are referring to flipping the graph upside down. Imagine taking each point on the graph and moving it to the other side of the x-axis. This kind of transformation is used to create the graph of \[y = -f(x)\]. Here, we change the sign of the output values of the function. That means, if you have a point \((x, y)\) on the original graph, after the transformation it turns into \((x, -y)\). In simpler terms, all the positive y-values become negative, and the negative y-values become positive.

• This transformation leaves the x-values unchanged.
• The effect is similar to "flipping over" so that what was above the x-axis is now below and vice versa.

This reflection affects the entire graph, impacting every point until the whole thing is mirrored. It's a great way to invert the behavior of a function over its respective range.

Reflection over the y-axis

Reflecting over the y-axis is a horizontal transformation of the graph that mirrors it across the y-axis. In terms of function transformation, this type of reflection is expressed as \[y = f(-x)\]. Unlike the reflection over the x-axis, here we are flipping the graph from left to right.

• The x-coordinate of each point is negated, turning a point \((x, y)\) into \((-x, y)\).
• This means if a part of the graph extends to the right of the y-axis, after reflection, it extends to the left, maintaining the same distance from the y-axis.

Simply put, the shape of the graph remains the same, but its position is horizontally reversed across the y-axis.

Graph Transformations

Graph transformations are operations that alter the position or the shape of a graph of a function. They include a variety of modifications such as translations, reflections, stretches, and compressions.In terms of reflections:

• Reflection over the x-axis transforms \(f(x)\) to \(-f(x)\), flipping the graph vertically.
• Reflection over the y-axis transforms \(f(x)\) to \(f(-x)\), flipping the graph horizontally.

Graph transformations help in understanding how complex functions behave based on simpler base functions like a basic line or a parabola. By visualizing these changes, one gains deep insights into the function's characteristics, such as symmetry, domain, and range.Comprehending these transformations not only makes graphing functions easier but also enhances problem-solving skills when analyzing how an equation or function behaves under different transformations.