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Chapter 1: Problem 16

, 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 is an ellipse with x-intercepts at \(x = 2 \pm \sqrt{2}\) and no real y-intercepts.

Step by step solution

01

Analyze the Equation for Symmetry

Initially, take the equation \( x^2 - 4x + 3y^2 = -2 \) and check for symmetry. Replace \(x\) with \(-x\) and \(y\) with \(-y\) one at a time to check for symmetry about the x-axis, y-axis, and origin. Neither substitution results in the original equation, indicating no symmetry.
02

Find x-intercepts

Set \(y = 0\) in the equation to find the x-intercepts. This gives \(x^2 - 4x = -2\). Solving for \(x\), rearrange to get \(x^2 - 4x + 2 = 0\). Using the quadratic formula \(x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\), we find the intercepts as \(x = 2 \pm \sqrt{2}\).
03

Find y-intercepts

Set \(x = 0\) in the equation to find the y-intercepts. We obtain \(3y^2 = -2 - (0)^2 + 4(0) = -2\). Since this results in imaginary roots, there are no real y-intercepts.
04

Plot Points and Draw the Graph

Choose points around the found x-intercepts, such as \( x = 0, 1, 3, \text{and } 4 \), substitute them in the equation to solve for corresponding values of \(y\). Use those points to sketch a rough graph. Note that the graph is an ellipse shifted along the x-axis.

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Key Concepts

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

X-Intercepts
To find the x-intercepts of an equation, you set the value of \( y \) to zero and solve for \( x \). This helps us identify where the graph crosses the x-axis. For the equation \( x^2 - 4x + 3y^2 = -2 \), when \( y = 0 \), the equation simplifies to \( x^2 - 4x = -2 \). Simply put, this becomes \( x^2 - 4x + 2 = 0 \).
  • Here, we need the roots of the quadratic equation \( x^2 - 4x + 2 = 0 \).
  • Once solved using the quadratic formula, we get \( x = 2 \pm \sqrt{2} \).
These values are the points where the graph intersects the x-axis. Always remember, x-intercepts are where the graph's height is zero.
Y-Intercepts
Y-intercepts occur where the graph crosses the y-axis, and you find these by setting \( x = 0 \) in the equation. For the equation \( x^2 - 4x + 3y^2 = -2 \), substituting \( x = 0 \) gives us a new equation: \( 3y^2 = -2 \). At this step, you would normally solve for \( y \), but here's the catch:
  • Since there's no real number that when squared gives a negative value, \( y \) becomes imaginary.
Thus, there are no real y-intercepts for this graph. This insight tells us the graph does not touch or cross the y-axis at any point.
Quadratic Formula
The quadratic formula is invaluable for solving equations of the form \( ax^2 + bx + c = 0 \). It allows us to find the roots of the equation, which are often the x-intercepts. The formula is: \[ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \]
  • Here, \( a \), \( b \), and \( c \) are the coefficients from your quadratic equation.
  • \( \pm \) indicates that there are two possible solutions: one using a plus and one using a minus.
Let's relate this back to our specific equation \( x^2 - 4x + 2 = 0 \):
  • \( a = 1 \), \( b = -4 \), and \( c = 2 \).
  • Substitute these values into the formula to get \( x = \frac{4 \pm \sqrt{16 - 8}}{2} \).
  • This simplifies to \( x = 2 \pm \sqrt{2} \).
The quadratic formula not only helps in finding intercepts but also aids in graph plotting by providing crucial points.
Symmetry Analysis
Symmetry analysis provides insights into the structure and behavior of graphing equations. Here's how it's typically done:
  • To find if a graph is symmetrical with respect to the x-axis, replace \( y \) with \( -y \) and see if the original equation remains unchanged.
  • For y-axis symmetry, substitute \( x \) with \( -x \).
  • If an equation maintains its form upon both substitutions, it has origin symmetry.
In our equation \( x^2 - 4x + 3y^2 = -2 \), replacing either \( x \) with \( -x \) or \( y \) with \( -y \) does not yield the original equation. Thus, there is no symmetry about the x-axis, y-axis, or origin. Knowing symmetry can save time when plotting because it helps predict the graph's shape by simply analyzing one part of it.

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