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Magazine Chapter 0: Problem 32
Plot the graph of the given equation. $$x=y^{2}+1$$
Short Answer
Expert verified
The graph is a parabola opening rightward with vertex at \((1, 0)\).
Step by step solution
01
Understanding the Equation
The given equation is \(x = y^{2} + 1\). Notice that this isn't in the standard form for linear, quadratic, or typical polynomial equations. Instead, we have \(x\) as a function of \(y\), suggesting a sideways parabola.
02
Identify the Shape and Vertex
Since the equation has the form \(x = y^{2} + 1\), it represents a parabola that opens horizontally in the positive \(x\)-direction. The vertex of this parabola is at the point \((1, 0)\), as it represents the minimum value of \(x\) when \(y = 0\).
03
Generate Points to Plot
To plot the parabola, substitute various values of \(y\) into the equation to find corresponding \(x\) values. For example, for \(y = 0\), \(x = 1\); for \(y = 1\), \(x = 2\); for \(y = -1\), \(x = 2\). Repeat this process for other values of \(y\) such as \(y = 2\) and \(-2\) to get \((5, \, 2)\) and \((5, \, -2)\).
04
Plot the Points on the Graph
Using the points found, plot them on a coordinate plane. Specifically, plot points like \((1, 0), (2, 1), (2, -1), (5, 2), (5, -2)\). These points should form the left side of the parabola, opening horizontally in the positive \(x\) direction.
05
Draw the Parabola
Connect the plotted points with a smooth curve that is symmetrical about the horizontal axis (x-axis). This curve should be parabolic, extending both upwards and downwards, towards positive \(x\) from the vertex \((1, 0)\).
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Key Concepts
These are the key concepts you need to understand to accurately answer the question.
Horizontal Parabola
In the realm of graphing, a horizontal parabola is a specific type of curve that opens sideways, unlike the more commonly discussed vertical parabolas. In a horizontal parabola, the independent variable is squared and appears in the form of an equation like \( x = y^2 + k \).
- **Orientation:** Horizontal parabolas open right or left, depending on the sign in front of the squared term.
- **Standard Form:** For horizontal parabolas, the equation typically looks like \( x = a(y - k)^2 + h \), where \( (h, k) \) is the vertex.
- **Characteristics:** Like their vertical counterparts, horizontal parabolas have axes of symmetry. However, in this case, the axis is horizontal, parallel to the x-axis.
Vertex of a Parabola
The vertex of a parabola is a fundamental concept as it represents the turning point of the curve. This is not limited to vertical parabolas but applies equally to horizontal ones. In the equation \( x = y^2 + 1 \), the vertex is found by identifying the values of \( h \) and \( k \) from the vertex form: \( x = a(y - k)^2 + h \). Here, it translates to the point \((1, 0)\).
- **Function:** The vertex acts as the minimal or maximal point for the parabola. In the case of \( x = y^2 + 1 \), it鈥檚 the point where \( x \) is at its minimum because it opens toward increasing \( x \) values.
- **Understanding:** Recognizing the vertex helps in plotting the curve accurately, as it determines the parabola's orientation and direction.
- **Application:** Knowing that for \( y = 0 \), \( x \) is at its minimum (1), we start plotting from this bottleneck, then move onwards.
Coordinate Plane
The coordinate plane serves as the canvas for graphing parabolas and other such functions. It's a two-dimensional space defined by a horizontal axis, the x-axis, and a vertical axis, the y-axis. This defined space enables us to plot and understand the relationships between different mathematical elements.
- **Axes:** The horizontal axis (x-axis) and vertical axis (y-axis) intersect at the origin point (0,0).
- **Quadrants:** The coordinate plane is divided into four quadrants, numbered counterclockwise. The signs of the coordinates for points in these quadrants vary, affecting the plot's visual structure.
- **Plotting Points:** For any mathematical graphing task, like plotting the equation \( x = y^2 + 1 \), it's key to identify and plot points first, then connect them. This involves substituting values to get ordered pairs like \((1, 0)\), \((2, 1)\), \((2, -1)\), and so on, determining their positions on the plane before shaping the curve.
