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Magazine Chapter 1: Problem 16
In Problems 1-30, plot the graph of each equation. Begin by checking for symmetries and be sure to find all \(x\) - and \(y\)-intercepts. $$ x^{2}-4 x+3 y^{2}=-2 $$
Short Answer
Expert verified
The graph has symmetry about the \(x\)-axis, with \(x\)-intercepts at \(x = 2 \pm \sqrt{2}\) and no real \(y\)-intercepts.
Step by step solution
01
Identify the Type of Equation
The given equation is \(x^2 - 4x + 3y^2 = -2\). This is a quadratic equation involving both \(x\) and \(y\), suggesting it might describe a conic section such as an ellipse or hyperbola. The negative constant indicates it's not a basic circle or ellipse.
02
Check for Symmetries
To check for symmetry, test the equation for changes when replacing \(x\) with \(-x\) and \(y\) with \(-y\). - Substituting \(-x\), the equation becomes \((-x)^2 - 4(-x) + 3y^2 = -2\), simplifying to \(x^2 + 4x + 3y^2 = -2\). This is not equivalent to the original equation, so there's no symmetry about the \(y\)-axis.- Substituting \(-y\), the equation becomes \(x^2 - 4x + 3(-y)^2 = -2\), which simplifies to the original equation \(x^2 - 4x + 3y^2 = -2\). This implies symmetry about the \(x\)-axis.
03
Find x-intercepts
Set \(y = 0\) to find \(x\)-intercepts:\[x^2 - 4x = -2\]Rearrange to get:\[x^2 - 4x + 2 = 0\]Use the quadratic formula, \(x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\), with \(a = 1, b = -4, c = 2\):\[x = \frac{4 \pm \sqrt{(-4)^2 - 4 \cdot 1 \cdot 2}}{2}\]\[x = \frac{4 \pm \sqrt{16 - 8}}{2}\]\[x = \frac{4 \pm \sqrt{8}}{2}\]\[x = \frac{4 \pm 2\sqrt{2}}{2}\]\[x = 2 \pm \sqrt{2}\]Thus, the \(x\)-intercepts are \(x = 2 + \sqrt{2}\) and \(x = 2 - \sqrt{2}\).
04
Find y-intercepts
Set \(x = 0\) to find \(y\)-intercepts:\[0^2 - 4(0) + 3y^2 = -2\]\[3y^2 = -2\]Divide both sides by 3:\[y^2 = -\frac{2}{3}\]Since \(y^2\) can't be negative for real numbers, there are no real \(y\)-intercepts.
05
Plot the Graph
Considering the obtained symmetries and intercepts:- Since there is symmetry about the \(x\)-axis, the graph will be symmetric above and below the \(x\)-axis.- The \(x\)-intercepts are at \(x = 2 + \sqrt{2}\) and \(x = 2 - \sqrt{2}\).- Since there are no real \(y\)-intercepts, the graph doesn't cross the \(y\)-axis.These suggest a hyperbolic or degenerate ellipse-like graph centered around some point in the \(x\) range derived from intercepts, but not stretching into the \(y\) axis.
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Key Concepts
These are the key concepts you need to understand to accurately answer the question.
Quadratic Equations
A quadratic equation is an important concept in algebra, characterized by its highest-degree term being squared. These equations typically come in the form of \( ax^2 + bx + c = 0 \), but can also involve two variables, as in the given exercise \( x^2 - 4x + 3y^2 = -2 \).
This particular form suggests a relation to conic sections because of the presence of both \(x^2\) and \(y^2\) terms. Conic sections are curves obtained from intersecting a plane with a double cone, and they include shapes like circles, parabolas, ellipses, and hyperbolas. Here, the negative constant suggests it is not a circular or an ellipse shape, hinting more towards a hyperbola. Understanding the format can give insights into the potential graph and behavior of the equation.
This particular form suggests a relation to conic sections because of the presence of both \(x^2\) and \(y^2\) terms. Conic sections are curves obtained from intersecting a plane with a double cone, and they include shapes like circles, parabolas, ellipses, and hyperbolas. Here, the negative constant suggests it is not a circular or an ellipse shape, hinting more towards a hyperbola. Understanding the format can give insights into the potential graph and behavior of the equation.
Symmetry in Graphs
Symmetry in graphs is a key concept that can simplify graphing and understanding equations significantly. In this case, we check for symmetrical properties by substituting \(-x\) and \(-y\) into the given equation.
- Replacing \( x \) with \(-x\) results in an equation \( x^2 + 4x + 3y^2 = -2 \) that is not identical to the original.
- For \(-y\), the equation returns to its original form \( x^2 - 4x + 3y^2 = -2 \), indicating symmetry about the \(x\)-axis.
Intercepts
Intercepts are critical for comprehending how a graph behaves. Identifying points where the graph cuts through the axes helps in plotting.
The \(x\)-intercepts are determined by setting \( y = 0 \), simplifying the equation to a single-variable quadratic equation: \( x^2 - 4x + 2 = 0 \). Solving this using the quadratic formula reveals the intercepts at \( x = 2 \pm \sqrt{2} \).
Finding the \(y\)-intercepts involves setting \( x = 0 \), resulting in \( 3y^2 = -2 \), which does not provide real number solutions. Hence, the graph does not intersect the \(y\)-axis. This tells us that the curve opens along the \(x\)-axis, reaffirming the symmetry.
The \(x\)-intercepts are determined by setting \( y = 0 \), simplifying the equation to a single-variable quadratic equation: \( x^2 - 4x + 2 = 0 \). Solving this using the quadratic formula reveals the intercepts at \( x = 2 \pm \sqrt{2} \).
Finding the \(y\)-intercepts involves setting \( x = 0 \), resulting in \( 3y^2 = -2 \), which does not provide real number solutions. Hence, the graph does not intersect the \(y\)-axis. This tells us that the curve opens along the \(x\)-axis, reaffirming the symmetry.
Graphing Techniques
Graphing is a skillful method of visually representing mathematical equations. To graph this particular equation \( x^2 - 4x + 3y^2 = -2 \), the collected information about symmetry and intercepts guides the process.
Start by plotting the \(x\)-intercepts at \( x = 2 \pm \sqrt{2} \), ensuring the graph does not cross the \(y\)-axis because this equation has no real \(y\)-intercepts.
The discovered symmetry about the \(x\)-axis means you'll need to reflect points across the axis to shape the full curve. By understanding these foundational graphing techniques, sketching the potential shape, whether it's a hyperbola-like structure or parts of a degenerate ellipse, becomes much more accessible. Utilizing technology or graph paper can assist in drafting a more precise representation.
Start by plotting the \(x\)-intercepts at \( x = 2 \pm \sqrt{2} \), ensuring the graph does not cross the \(y\)-axis because this equation has no real \(y\)-intercepts.
The discovered symmetry about the \(x\)-axis means you'll need to reflect points across the axis to shape the full curve. By understanding these foundational graphing techniques, sketching the potential shape, whether it's a hyperbola-like structure or parts of a degenerate ellipse, becomes much more accessible. Utilizing technology or graph paper can assist in drafting a more precise representation.
