Question

If (3,6) is a point on the graph of \(y=f(x),\) which of the following points must be on the graph of \(y=-f(x) ?\) (a) (6,3) (b) (6,-3) (c) (3,-6) (d) (-3,6)

Short answer

(c) (3, -6)

Step-by-step solution

- Understand the Transformation

The function transformation from \(y = f(x)\) to \(y = -f(x)\) reflects the graph over the x-axis. This means that for any point \((a, b)\) on the graph of \(y = f(x)\), the corresponding point \((a, -b)\) will be on the graph of \(y = -f(x)\).

- Find the Reflected Point

Given the point \((3, 6)\) on the graph of \(y = f(x)\), reflect this point over the x-axis. The new point will be \((3, -6)\) since we change the y-coordinate from positive 6 to negative 6.

- Identify the Correct Option

From the provided options, find the point that matches the reflected point \((3, -6)\). The correct option is (c) \((3, -6)\).

Key concepts

Graph Reflection

Graph reflection is a type of transformation where the graph of a function is flipped over a line, known as the axis of reflection. In coordinate geometry, this can be either the x-axis, y-axis, or any other line.
For example, if you have a point (x, y) on the original graph, and you reflect it over the x-axis, the new point will be (x, -y).
Reflections help us understand how changing the sign of coordinates affects the position of points on the plane.

X-Axis Reflection

An x-axis reflection flips the graph of a function over the x-axis. This means that every point on the graph of the original function will be reflected vertically.
For instance, if the point (3, 6) is on the graph of the function y = f(x), reflecting it over the x-axis will give you the point (3, -6) on the graph of y = -f(x).
To generalize:

• If the original point is (x, y), the reflected point will be (x, -y).

This is very useful in understanding how negative signs affect the output values of a function.

Function Transformation

Function transformation refers to changes made to the graph of a function. These changes can include translations, reflections, stretches, and compressions.
A common transformation is changing y = f(x) to y = -f(x), which reflects the graph over the x-axis as discussed above.
These transformations are essential in graphing functions and understanding their behavior. Here’s a quick overview:

• Translation: Shifting the graph horizontally or vertically.
• Reflection: Flipping the graph over an axis.
• Stretching/Compressing: Changing the graph's scale along an axis.

Each transformation helps visualize how different operations affect the graph of a function.

Coordinate Geometry

Coordinate geometry is the study of geometric figures using a coordinate system. It uses algebraic equations to represent geometric shapes and their transformations.
Points, lines, and curves are all placed on the coordinate plane. Every point is defined by a pair of coordinates (x, y).
Understanding this helps in visualizing transformations like reflections.

• If you reflect over the x-axis, you change the y-coordinate (from y to -y).
• If you reflect over the y-axis, you change the x-coordinate (from x to -x).

Mastering these concepts allows for a deep understanding of how function transformations work in a coordinate plane.