Problem 7 - Elementary and Intermediate Algebra

Question

Fill in the blank. The graph of \(g(x)=-x^{2}\) is the _____ of the graph of \(f(x)=x^{2}\) about the \(x\) -axis.

Step-by-Step Solution

Answer

reflection

Step 1: Understand the Graphs

First, note that the graph of the function \( f(x) = x^2 \) is a parabola facing upwards with its vertex at the origin \((0,0)\).

Step 2: Identify the Transformation

The graph of the function \( g(x) = -x^2 \) involves multiplying the function \( f(x) \) by \(-1\). This transformation affects the direction of the parabola, as multiplying by \(-1\) will reflect the graph across the \(x\)-axis.

Step 3: Understand Reflection

Reflection of a graph across the \(x\)-axis involves changing the sign of the \(y\)-values. For the graph of \( f(x) \), which has positive \(y\)-values, reflecting about the \(x\)-axis will make all \(y\)-values negative, resulting in the graph of \( g(x) = -x^2 \) being a downward-facing parabola.

Step 4: Confirmation

Both the original function \( f(x) = x^2 \) and the transformed function \( g(x) = -x^2 \) are parabolas, but they face opposite directions due to the reflection about the \(x\)-axis. This makes \( g(x) \) the reflection of \( f(x) \).

Key Concepts

Transformation

The concept of transformation in mathematics refers to various operations that can be performed on functions to alter their graphs. In this context, you can think of a transformation as a way to "move" or "change" a graph in a specific manner. The transformation applied to the function in our exercise is a reflection.

A **reflection** is a type of rigid transformation, and it flips the graph of a function over a specified line. For example, in the exercise, the line of reflection is the \(x\)-axis. When you reflect a graph across this line, you effectively change the sign of all the \(y\)-coordinates of the points on the graph.

The transformation performed involves multiplying the function by \(-1\). This operation changes the direction in which the parabola opens. Before the transformation, the function \( f(x) = x^2 \) results in an upward opening parabola, and after applying the transformation, the graph of \( g(x) = -x^2 \) opens downward as it flips over the \(x\)-axis.

Parabola

A parabola is a symmetrical, U-shaped curve that you often encounter in graphing quadratic functions. Its standard form is \( f(x) = ax^2 + bx + c \). In the context of this exercise, we deal specifically with simple parabolas, \( f(x) = x^2 \) and \( g(x) = -x^2 \).

Both of these functions demonstrate the basic form of a parabola world:

• In \( f(x) = x^2 \), the parabola opens upwards, reflecting that the coefficient of \(x^2\) is positive.
• In \( g(x) = -x^2 \), the parabola opens downward, indicating the coefficient is negative.

The vertex of a parabola represents its highest or lowest point. For these functions, the vertex is located at the origin, \((0, 0)\). By reflecting \( f(x) = x^2 \), its upward facing parabola flips to a downward facing one as seen in \( g(x) = -x^2 \).

Understanding parabolas is essential as they are a fundamental aspect of quadratic functions, and their transformations help in visualizing the changes to the function's graph.

Graph of a Function

The graph of a function shows a visual representation of all the solutions for an equation in the coordinate plane. Every point on the graph corresponds to an \(x\) and \(y\) coordinate that satisfies the function's equation.

For the functions \( f(x) = x^2 \) and \( g(x) = -x^2 \), these graphs show parabolas that are transformations of each other.

• \( f(x) = x^2 \) is a simple upward-facing parabola, representing the set of all \((x, y)\) pairs where \(y = x^2\).
• \( g(x) = -x^2 \) is its downward-facing counterpart where \(y = -x^2\).

When drawing these graphs:

• The vertex at the origin remains fixed.
• The direction of the parabola helps identify whether the graph is reflected or not.

Visualizing these graphs strengthens the comprehension of how different transformations like reflection impact the entire set of solutions visually. Graphs are invaluable for interpreting functions and their transformations.