Question

The graph of a function \(y=f(x)\) is contained completely in the fourth quadrant. Copy and complete each statement. a) If \(y=f(x)\) is transformed to \(y=-f(x)\) it will be in quadrant b) If \(y=f(x)\) is transformed to \(y=f(-x)\) it will be in quadrant c) If \(y=f(x)\) is transformed to \(y=4 f(x)\) it will be in quadrant d) If \(y=f(x)\) is transformed to \(y=f\left(\frac{1}{4} x\right)\) it will be in quadrant

Short answer

a) First quadrant, b) Third quadrant, c) Fourth quadrant, d) Fourth quadrant

Step-by-step solution

Define the fourth quadrant

In the coordinate plane, the fourth quadrant is the region where the x-values are positive and the y-values are negative.

Analyze the transformation for part (a)

The transformation from \(y=f(x)\) to \(y=-f(x)\) reflects the graph over the x-axis. Since the original graph is in the fourth quadrant with positive x-values and negative y-values, reflecting over the x-axis will switch the negative y-values to positive y-values. Thus, the graph will be in the first quadrant.

Analyze the transformation for part (b)

The transformation from \(y=f(x)\) to \(y=f(-x)\) reflects the graph over the y-axis. Since the original graph is in the fourth quadrant with positive x-values and negative y-values, reflecting over the y-axis will switch the positive x-values to negative x-values. Therefore, the graph will be in the third quadrant.

Analyze the transformation for part (c)

The transformation from \(y=f(x)\) to \(y=4f(x)\) stretches the graph vertically by a factor of 4. It does not change the signs of the x or y-values, so the graph remains in the fourth quadrant.

Analyze the transformation for part (d)

The transformation from \(y=f(x)\) to \(y=f(\frac{1}{4} x)\) stretches the graph horizontally by a factor of 4. It does not change the signs of the x or y-values, so the graph remains in the fourth quadrant.

Key concepts

Coordinate Plane

The coordinate plane is a two-dimensional space divided by two perpendicular lines, known as the x-axis (horizontal line) and the y-axis (vertical line). The point where these lines intersect is called the origin, denoted as (0, 0). This plane is used to graph functions and points, which are identified by coordinates \((x, y)\).

Coordinates are written as ordered pairs, with the first number representing the horizontal position (x-axis) and the second representing the vertical position (y-axis). For example, the point (3, -2) is 3 units to the right of the origin and 2 units down.

Understanding the coordinate plane is crucial for analyzing graph transformations, as it allows us to visualize how a function changes position or shape after a transformation.

Quadrants

The coordinate plane is divided into four sections known as quadrants. These quadrants help in identifying the location of points and the behavior of functions.

The quadrants are:

• First Quadrant: Top-right section where both x and y values are positive (x > 0, y > 0).
• Second Quadrant: Top-left section where x values are negative and y values are positive (x < 0, y > 0).
• Third Quadrant: Bottom-left section where both x and y values are negative (x < 0, y < 0).
• Fourth Quadrant: Bottom-right section where x values are positive and y values are negative (x > 0, y < 0).

For example, if a graph is contained completely in the fourth quadrant, then every point on the graph will have positive x-values and negative y-values. Transformations that reflect the graph about the axes or stretch/compress the graph can change which quadrant the graph lies in.

Function Transformations

Function transformations alter the position or shape of a graph. Understanding these shifts, reflections, and stretches/compressions is key to mastering graph transformation problems. Here are some common transformations:

• Reflections: Reflecting a function over the x-axis changes its sign. For a function \(y = f(x)\), the transformed function \(y = -f(x)\) will reflect the graph over the x-axis.
• Reflections about the y-axis: For \(y = f(-x)\), this reflects the graph over the y-axis.
• Vertical Stretch/Compression: Multiplying a function by a constant \(a\) (say, \(y = af(x)\)) will stretch (if \(a > 1\)) or compress (if \(0 < a < 1\)) the graph vertically.
• Horizontal Stretch/Compression: For \(y = f(ax)\), dividing x by \(a\) will horizontally stretch or compress the graph. If \(0 < a < 1\), the graph stretches, and if \(a > 1\), the graph compresses.

These transformations are critical to understanding how a graph's shape and position change. When dealing with transformations, always pay attention to changes in the x and y values, as they dictate which quadrant the transformed graph will occupy.