Problem 57 - Beginning and Intermediate Algebra

Question

If the following transformations are performed on the graph of \(f(x)\) to obtain the graph of \(g(x),\) write the equation of \(g(x)\). \(f(x)=x^{2}\) is reflected about the \(x\) -axis.

Step-by-Step Solution

Answer

The equation of g(x) is \(g(x) = -x^2\), which is the reflection of f(x) = x^2 over the x-axis.

Step 1: Understand the reflection over the x-axis #

To reflect a function over the x-axis means that each point (x, f(x)) on the graph of f will be transformed to a point (x, -f(x)) on the graph of g. In other words, the new function g(x) will be the negation of f(x) at each x-value.

Step 2: Write the equation for g(x) #

Recall that f(x) = x^2. To find g(x), we need to negate the function f(x) at each x-value. So, g(x) = -f(x) which gives us: g(x) = -x^2 Thus, the equation of g(x) is \(g(x) = -x^2\).

Key Concepts

Graph Transformations

Graph transformations involve changing the position or shape of a graph in a coordinate plane. These transformations do not completely alter the nature or the properties of the graph itself. Instead, they help us visualize how modifying a function's equation impacts its appearance. Some common transformations include:

• Vertical and horizontal shifts: moving the graph up/down or left/right.
• Scaling: stretching or compressing the graph in either the vertical or horizontal direction.
• Reflection: flipping the graph over a specific axis.
• Rotations: changing the graph's orientation in the plane.

Each transformation has its unique impact. A graph of a quadratic function, such as a parabola, can easily demonstrate these effects.
For instance, when applying a reflection over the x-axis, each point on the parabola gets flipped along the horizontal axis, resulting in an upside-down shape.

Reflection Over the X-axis

Reflecting a function over the x-axis is a specific type of graph transformation. This process involves taking each point on the original graph and flipping it vertically to create a mirror image along the x-axis.
In technical terms, if you have a function denoted by \( f(x) \), then its reflection over the x-axis is given by \( g(x) = -f(x) \). You'll negate the y-values of all points while keeping the x-values unchanged.

Reflecting the quadratic function \( f(x) = x^2 \) over the x-axis transforms it into \( g(x) = -x^2 \). This means that the entire parabola, which originally opens upwards, will now open downwards. Every point that was originally above the x-axis is now equally below it, giving a complete flip to the graph's orientation.

Function Notation

Function notation is a way to name and define a function using variables and expressions. It's a streamlined approach used in mathematics to express the input-output relationship clearly.

• Typically, a function is written as \( f(x) \), where \( f \) names the function, and \( x \) represents the input variable.
• The expression that follows the equal sign shows how to compute the function's output for any given \( x \).
• For instance, \( f(x) = x^2 \) tells us that the function squares the input \( x \) to produce the output.

This notation is crucial when performing transformations. Consider when a reflection over the x-axis is applied: clearly use function notation to show this as \( g(x) = -f(x) \). It succinctly captures the transformation by highlighting the change from \( f(x) \) to \( g(x) \). Such notation is indispensable for expressing complex mathematical operations efficiently.