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Chapter 0: Problem 16

Graph. (Unless directed otherwise, assume that "Graph" means "Graph by hand.") \(x=2-y^{2}\)

Short Answer

Expert verified
The parabola \( x = 2 - y^2 \) opens to the left with a vertex at (2, 0).

Step by step solution

01

Identify the equation type

The given equation \(x = 2 - y^2\) is a sideways parabola. This can be rearranged to the standard form of a sideways parabola \(x = a(y - k)^2 + h\), where \(a = -1\), \(h = 2\), and \(k = 0\). The graph opens to the left because \(a\) is negative.
02

Determine key features

Since the equation is in the form \(x = 2 - y^2\), it has a vertex at \((2, 0)\). The parabola opens to the left, implying it will curve toward negative values on the x-axis. The axis of symmetry is the horizontal line \(y = 0\).
03

Calculate additional points

Choose a few values for \(y\) to calculate corresponding \(x\)-coordinates. For example, if \(y = 1\), then \(x = 2 - 1^2 = 1\). If \(y = -1\), then \(x = 2 - (-1)^2 = 1\). Similarly, for \(y = 2\) or \(y = -2\), \(x = 2 - 4 = -2\). Calculate more points if necessary to ensure an accurate graph.
04

Plot the points

Plot the vertex \((2, 0)\) on the coordinate plane. Then, plot the points \((1, 1)\), \((1, -1)\), \((-2, 2)\), and \((-2, -2)\). Ensure these points reflect the parabola opening to the left.
05

Draw the curve

Draw a smooth curve through the points plotted, ensuring the curvature follows the direction of a left-opening parabola. Make sure the curve is symmetrical about the line \(y = 0\).

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Key Concepts

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

Sideways Parabola
A sideways parabola is a parabola that opens horizontally instead of vertically. Unlike the traditional parabolas that open upwards or downwards, a sideways parabola points left or right on the graph. The equation typically takes the form \( x = a(y - k)^2 + h \), where \( a \) determines the direction of opening:
  • If \( a > 0 \), the parabola opens to the right.
  • If \( a < 0 \), the parabola opens to the left.
For instance, in the equation \( x = 2 - y^2 \), the parabola is sideways because the \( y^2 \) term is isolated on one side. Here, \( a = -1 \), signaling the parabola opens to the left. Sideways parabolas are particularly useful in certain fields of engineering and physics due to their unique orientation.
Vertex
The vertex of a parabola represents its highest or lowest point, depending on its orientation. In the context of a sideways parabola, the vertex is the farthest left or right point:
  • For the equation \( x = 2 - y^2 \), the vertex is located at the point \((h, k) = (2, 0)\).
This point marks the tip of the curve where the direction changes. Understanding the location of the vertex helps in plotting the parabola accurately. The vertex is crucial because it provides a starting point for sketching the graph and determining the parabola's extension direction. It is also where the axis of symmetry intersects.
Axis of Symmetry
The axis of symmetry is an imaginary line that vertically or horizontally splits the parabola into two mirror-image halves. For a sideways parabola, this axis is horizontal rather than vertical.
  • The equation \( y = 0 \) serves as the axis of symmetry for the parabola \( x = 2 - y^2 \).
This line is helpful as it ensures that for any given \( y \), the \( x \)-coordinates on either side of the axis are equidistant from each other. It simplifies the creation of additional graph points because it guarantees the parabola's symmetry. Consequently, plotting points on one side will mirror those on the opposite side, speeding up the graphing process. Being aware of the axis of symmetry also aids in confirming the accuracy of the drawn parabola.
Plotting Points
Plotting points is a practical step to verify the shape and direction of a parabola on a graph. For the equation \( x = 2 - y^2 \), calculating a couple of additional points besides the vertex makes it easier to draw the parabola:
  • By substituting \( y = 1 \), we find \( x = 1 \), giving the point \((1, 1)\).
  • For \( y = -1 \), \( x = 1 \), resulting in \((1, -1)\).
  • When \( y = 2 \) or \( y = -2 \), \( x = -2 \), leading to points \((-2, 2)\) and \((-2, -2)\).
Plotting these points on a graph provides a framework for the parabola. Ensuring equal distances from the axis of symmetry will result in an even curve. Each point confirms the leftward opening and correctness of the plotting, building toward the final, smooth curve that represents the sideways parabola accurately.

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