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Chapter 8: Problem 33

How does the graph of \(-y=x^{2}\) compare to the graph of \(y=x^{2}\) ? Explain your answer.

Short Answer

Expert verified
The graph of \( y = -x^2 \) flips the parabola upside down compared to \( y = x^2 \).

Step by step solution

01

Understand the Original Graph

The graph of \( y = x^2 \) is a parabola that opens upwards with its vertex at the origin \((0, 0)\). The parabola is symmetric around the y-axis.
02

Analyze the Effect of the Negative Sign

The equation \( -y = x^2 \) can be rewritten as \( y = -x^2 \) by multiplying both sides by -1. This transformation changes the direction of the parabola from opening upwards to opening downwards.
03

Compare the Graphs

In \( y = x^2 \), the parabola opens upwards, while in \( y = -x^2 \), the parabola opens downwards but still retains the same vertex \((0, 0)\) and symmetry around the y-axis.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Graph Transformations
Graph transformations involve changing the appearance or position of a graph without altering its fundamental shape. When you transform the equation \(y = x^2\) into \(y = -x^2\), you alter how the parabola opens. Multiplying the entire function by -1 is a vertical reflection. This specific transformation flips the graph over the x-axis. Therefore, a parabola that opens upwards now opens downwards. The shape remains the same; only its orientation changes. Graph transformations allow us to manipulate graphs to fit specific models or data representations more effectively.
Parabolas
A parabola is a U-shaped curve that can open upwards or downwards in its basic form. The general equation of a parabola is \(y = ax^2 + bx + c\). In the simplest form, the parabola \(y = x^2\) opens upwards. It represents one of the most fundamental quadratic functions. Parabolas have unique properties like a vertex and axis of symmetry. They appear naturally in physics and engineering, modeling scenarios like the path of a thrown ball or the cross-section of a satellite dish. Understanding parabolas helps in solving problems related to quadratic equations and analyzing their graphs.
Vertex (0,0)
The vertex is a crucial point on a parabola that represents its maximum or minimum point. For the quadratic \(y = x^2\) or \(y = -x^2\), the vertex is located at the origin, \((0,0)\). Here, the parabola changes direction. It is the point of symmetry, and it divides the parabola into two mirrored halves. In the case of \(y = x^2\), the point \((0,0)\) is the lowest point, or the minimum. For \(y = -x^2\), it becomes the highest point, or the maximum. Understanding the vertex helps in graphing parabolas and identifying key features of their curves.
Symmetry around the y-axis
Symmetry around the y-axis means if you were to fold the graph along the y-axis, both sides would match perfectly. Parabolas described by functions such as \(y = x^2\) or \(y = -x^2\) showcase this symmetry. The y-axis acts as a natural mirror, ensuring one side of the graph is a reflection of the other. This property simplifies the analysis and graphing of quadratic functions. Since both forms of the equation maintain this symmetry, it is a useful aspect when comparing or contrasting different quadratic graphs.

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Most popular questions from this chapter

\(x^{2}+y^{2}-16 x+6 y+71=0\)

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