Problem 46 - College Algebra with Modeling and Visualization

Question

Use transformations to sketch a graph of \(f\). \(f(x)=-x^{2}\)

Step-by-Step Solution

Answer

Reflect \(y = x^2\) over the x-axis to get \(f(x) = -x^2\), a downward-opening parabola.

Step 1: Identify the Parent Function

The parent function in this case is the simplest form of a quadratic function, which is \(y = x^2\). This is a standard parabola opening upwards with its vertex at the origin (0,0).

Step 2: Apply the Reflection Transformation

The given function is \(f(x) = -x^2\). The negative sign in front of \(x^2\) indicates a reflection across the x-axis. This means every point of the parent function \(y = x^2\) is reflected downwards. As a result, the parabola opens downwards instead of upwards.

Step 3: Sketch the Transformed Graph

After reflecting the parent function \(y = x^2\) over the x-axis, sketch the graph. It will have the same vertex at the origin (0, 0), but the parabola will open downwards. The arms of the parabola will approach negative infinity as \(x\) moves towards positive or negative infinity. The axis of symmetry remains the y-axis (or \(x = 0\)).

Key Concepts

Quadratic Functions

Quadratic functions are a type of polynomial function, distinguished by their highest power of the variable being two. They take the general form of \( f(x) = ax^2 + bx + c \), where \(a\), \(b\), and \(c\) are constants. A key feature of quadratic functions is their graph, known as a parabola. A standard quadratic function \( y = x^2 \) creates a symmetrical parabola centered around the vertical line called the axis of symmetry, usually at \( x = 0 \). The lowest or highest point on a parabola (depending on whether it opens upwards or downwards) is known as the vertex. For \( y = x^2 \), the vertex is at the origin \((0,0)\). Since quadratic functions are fundamental to algebra, understanding their properties and transformations is crucial for solving a wide range of problems.

Parabola Reflection

Reflecting a parabola across the x-axis is a common transformation in algebra. In the case of quadratic functions, this transformation is indicated by a negative sign in front of the \( x^2 \) term. For example, the function \( f(x) = -x^2 \) is derived from the parent function \( y = x^2 \) by reflecting across the x-axis. This mathematical operation flips the parabola, so it opens downwards instead of upwards.

• The vertex of the parabola remains unchanged at the origin \((0,0)\).
• The axis of symmetry also remains the same, which is the y-axis or \( x = 0 \).
• Each point of the original parabola \( y = x^2 \) is mirrored directly below the x-axis.

This transformation is useful for graphing and solving real-world problems where the context demands an inversely proportional relationship.

Graph Sketching Techniques

Graph sketching involves careful analysis and application of transformations to understand the behavior of a function visually. One starts by identifying basic properties like the vertex, axis of symmetry, and direction of opening. To sketch \( f(x) = -x^2 \), begin by considering the parent graph \( y = x^2 \):

• The function's vertex is at the origin \((0,0)\).
• The axis of symmetry is the vertical line through the vertex, \( x = 0 \).
• The parabola, due to reflection, opens in the opposite direction from \( y = x^2 \), namely downwards.

As \( x \) moves away from the origin in either direction, the values of \( f(x) \) decrease, unlike the upward-opening parent parabola. Consider tracking a few key points such as \((1,-1)\) and \((-1,-1)\), illustrating the downward opening by plotting these on the coordinate plane. This methodical approach helps someone visualize and plot the transformed graph correctly.