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

Option (d), the point (-3,6).

Step-by-step solution

Identify the given point

The given point is (3,6) and lies on the graph of the function \(y = f(x)\). This means that when \(x = 3\), \(y = 6\), so \(f(3) = 6\).

Understand the transformation

When given the transformed function \(y = f(-x)\), we substitute \(-x\) for \(x\) in the original function. This means for the transformed function, \(f(-x)\) will be equal to \(y\).

Determine the equivalent point for the transformed function

Given \(f(3) = 6\), we need to find the corresponding point on the function \(y = f(-x)\). Since \(y = 6\) when \(3 = -x\), solving for \(x\), we get \(x = -3\).

Identify the coordinates of the new point

The new point on the graph of \(y = f(-x)\) can be written as (-3,6) because \(f(-(-3)) = f(3) = 6\).

Choose the correct option

The point (-3,6) corresponds to option (d) in the given choices.

Key concepts

Graph of a Function

The graph of a function is an instrumental concept in mathematics. It's a visual representation of all the ordered pairs \(x, y\) that satisfy the function \(y = f(x)\). Each point on the graph shows a unique relationship between an input \(x\) and its corresponding output \(y\).
For example, if we have the point (3,6) on the graph of \(y = f(x)\), it tells us that when \(x = 3\), the function gives us \(y = 6\). This point plays a crucial role when examining transformations and symmetries of functions.

Transformations of Functions

Transforming functions is about changing their input or output values in a systematic way. Let's explore how this works using the exercise provided.
\(y = f(-x)\) implies a horizontal reflection of the original function's graph across the y-axis. To convert a point from the graph of \(y = f(x)\) to the graph of \(y = f(-x)\), we simply negate the \(x\)-coordinate.
For instance, the point (3, 6) on \(y = f(x)\) maps to (-3, 6) on \(y = f(-x)\). Here’s a quick reference for some common transformations:

• Horizontal shifts: \(y = f(x \pm h)\)
• Vertical shifts: \(y = f(x) \pm k\)
• Reflections: \(y = f(-x)\) or \(y = -f(x)\)
• Stretches and Compressions: \(y = af(x)\) or \(y = f(bx)\)

Symmetry in Functions

Symmetry in functions is a beautiful and fundamental property. Certain functions have symmetrical graphs that make them easier to understand and analyze. There are two main types of symmetry in functions:
\bullet\ **Even Functions:** A function is even if \(f(x) = f(-x)\) for all \(x\) in its domain. The graph of an even function is symmetric with respect to the y-axis. For instance, \(f(x) = x^2\) is an even function because its graph looks the same on either side of the y-axis.
\bullet\ **Odd Functions:** A function is odd if \(f(-x) = -f(x)\) for all \(x\) in its domain. The graph of an odd function is symmetric with respect to the origin. For example, \(f(x) = x^3\) is an odd function since rotating its graph 180 degrees around the origin would yield the same graph.
In our example, examining the graph of \(y = f(-x)\) aligned with an understanding of these symmetries further solidifies the knowledge of how functions transform visually.