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Chapter 0: Problem 2

Plot the graph of each equation. Begin by checking for symmetries and be sure to find all \(x\) - and \(y\) -intercepts. \(x=-y^{2}+1\)

Short Answer

Expert verified
The graph is a sideways parabola symmetric about the x-axis with vertex at (1, 0).

Step by step solution

01

Check for Symmetry

To check for symmetry, we explore symmetrical properties through two tests: reflection over the y-axis, and origin.1. **Reflection over x-axis:** Replace \(y\) with \(-y\), the equation becomes \(x = -(-y)^2 + 1\), which simplifies to \(x = -y^2 + 1\). Since we obtain the same equation, the graph is symmetric about the x-axis.2. **Reflection over y-axis**: Substitute \(x\) with \(-x\), leading to the equation \(-x = -y^2 + 1\), different from the original, thus no symmetry about the y-axis.3. **Symmetry about the origin:** Replace \(x\) with \(-x\) and \(y\) with \(-y\): \(-x = -(-y)^2 + 1\), which simplifies to \(-x = -y^2 + 1\), still different from the original equation, so no origin symmetry.Conclusion: The graph is symmetric about the x-axis.
02

Find the x-Intercepts

To find the x-intercepts, set \(y = 0\) in the equation \(x = -y^2 + 1\).Substitute: \[ x = -(0)^2 + 1 = 1 \].The x-intercept is \((1, 0)\).
03

Find the y-Intercepts

For y-intercepts, set \(x = 0\): \[ 0 = -y^2 + 1 \] which implies \[-y^2 = -1\] and \(y^2 = 1\).Take the square root: \(y = \pm 1\).The y-intercepts are \((0, 1)\) and \((0, -1)\).
04

Plot Key Points and Graph

1. **Identify the vertex**: Since the equation is in the form \(x = -y^2 + 1\), it's a sideways parabola opening to the left. The vertex, where the value of \(y = 0\), gives \(x=1\), so the vertex is \((1, 0)\).2. **Plot Intercepts**: Plot intercepts from Steps 2 and 3: \((1, 0)\), \((0, 1)\), and \((0, -1)\).3. **Sketch Graph**: Draw a parabola passing through these points and opening to the left.

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Key Concepts

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

Symmetry in Graphs
Symmetry in graphs means that a graph is a mirror image about a particular axis or center. Understanding symmetry helps us sketch graphs more accurately and efficiently. There are three main types of symmetry to check in an equation:
  • Symmetry about the y-axis: To check, replace every "x" in the equation with "-x". The resulting equation is compared to the original. If they match, the graph is symmetric about the y-axis.
  • Symmetry about the x-axis: Here, replace "y" with "-y" and compare the equations. If the equations are equivalent, the graph is symmetric about the x-axis.
  • Symmetry about the origin: Substitute both "x" with "-x" and "y" with "-y". If the resulting equation is the same as the original, the graph is symmetric about the origin.
In the equation from the exercise, we found symmetry about the x-axis only. This tells us that the graph can be flipped over the x-axis without any change in shape.
Intercepts
Intercepts are crucial points on a graph where it crosses the x-axis or y-axis. These points are often used as landmarks to aid in graphing. To find intercepts:
  • x-intercepts: Set y = 0 in the equation and solve for x. This will show where the graph intersects the x-axis. For instance, in the given equation, when y = 0, solving gives x = 1, meaning the x-intercept is at the point (1, 0).
  • y-intercepts: Set x = 0 and solve for y. This is where the graph crosses the y-axis. Solving the example equation for x = 0 results in y-values of 1 and -1, giving y-intercepts at (0, 1) and (0, -1).
Finding and plotting these intercepts helps in sketching the basic outline of the graph, providing guides on where the curve should pass.
Parabolas
A parabola is a U-shaped curve, which in some equations, can appear sideways like in our exercise. The general form of a sideways parabola can be expressed as \[ x = a(y - k)^2 + h, \]where (h, k) is the vertex of the parabola.In the exercise equation \[ x = -y^2 + 1, \]we identify a sideways parabola opening to the left since "a" is negative.

Key Features of a Sideways Parabola:

  • Vertex: The highest or lowest point on the curve. For this equation, since y = 0 gives the maximum x-value of 1, the vertex is (1, 0).
  • Opening direction: Determined by the sign of "a". A negative "a" implies the parabola opens to the left, and a positive "a" would open to the right.
  • Axis of symmetry: A line passing through the vertex, bisecting the parabola into two mirror-image sections. For a sideways parabola, this is a horizontal line parallel to the x-axis.
These features are essential for drawing and understanding the behavior of parabolas in equations, ensuring the graph is plotted accurately.

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