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Chapter 1: Problem 21

Graph the equation \(4 x^{2}+2 y^{2}=1\) by solving for \(y\) and graphing two equations corresponding to the negative and positive square roots. (This graph is called an ellipse.)

Short Answer

Expert verified
Graph \(y = \pm \sqrt{\frac{1 - 4x^2}{2}}\) to form an ellipse.

Step by step solution

01

Recognize the Standard Form of an Ellipse

The given equation \(4x^2 + 2y^2 = 1\) is in the form of an ellipse equation. An ellipse generally comes in the form \(\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1\). Here, we need to manipulate the given equation into this form to solve for \(y\).
02

Rearrange the Equation

We start by isolating the \(y^2\) term. Divide every term in the equation \(4x^2 + 2y^2 = 1\) by 1 to make the right side equal to 1: \[\frac{4x^2}{1} + \frac{2y^2}{1} = 1\] This is the equation of an ellipse with modifications.
03

Solve for \(y^2\)

Isolate \(y^2\) by moving \(4x^2\) to the other side: \[2y^2 = 1 - 4x^2\] Now, solve for \(y^2\) by dividing through by 2:\[y^2 = \frac{1 - 4x^2}{2}\]
04

Solve for \(y\)

Take the square root of both sides to solve for \(y\). Remember to consider both the positive and negative square roots:\[y = \pm \sqrt{\frac{1 - 4x^2}{2}}\]This gives us two equations: \(y = \sqrt{\frac{1 - 4x^2}{2}}\) and \(y = -\sqrt{\frac{1 - 4x^2}{2}}\).
05

Graph the Equations

Graph \(y = \sqrt{\frac{1 - 4x^2}{2}}\) for the top half of the ellipse, and \(y = -\sqrt{\frac{1 - 4x^2}{2}}\) for the bottom half of the ellipse. You can use a graphing calculator or software to plot these two functions and observe the complete ellipse.

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Key Concepts

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

Standard Form of an Ellipse
An ellipse is a fascinating shape in geometry that resembles a flattened circle. Its standard form helps us understand and describe the ellipse mathematically. Generally, an ellipse is represented by the equation:
  • \[ \frac{x^2}{a^2} + \frac{y^2}{b^2} = 1 \]
In this equation, \(a\) and \(b\) are constants that determine the shape and size of the ellipse:
  • \(a^2\) is associated with the \(x\)-axis, and \(b^2\) is associated with the \(y\)-axis.
  • The values of \(a\) and \(b\) depict the semi-major and semi-minor axes, respectively.
To convert an equation like \(4x^2 + 2y^2 = 1\) into standard form, we need to reorganize it to match \( \frac{x^2}{a^2} + \frac{y^2}{b^2} = 1 \). This involves adjusting the coefficients of \(x^2\) and \(y^2\) so that they represent the squares of \(a\) and \(b\). This step is crucial before solving further into other steps such as graphing.
Solving for y in Equations
When you have an equation involving \(y\), sometimes it's necessary to solve for \(y\) to understand or graph the function. Our equation \(4x^2 + 2y^2 = 1\) requires us to isolate \(y\).^2. Initially, rearrange it to focus on the \(y^2\)-term:
  • Subtract \(4x^2\) from both sides: \(2y^2 = 1 - 4x^2\).
  • Divide everything by 2 to solve for \(y^2\): \(y^2 = \frac{1 - 4x^2}{2}\).
Now, to obtain \(y\), take the square root of both sides:
  • The square root introduces a \(\pm\) symbol, representing both positive and negative roots.
  • This means we have two expressions for \(y\): \(y = \pm \sqrt{\frac{1 - 4x^2}{2}}\).
This solution accounts for the full range of \(y\) values represented in the ellipse's graph, giving us both the upper and lower sides.
Graphing Equations
Graphing an equation involves plotting points or lines on a coordinate plane to visually represent the relationship between variables. In the case of an ellipse, once you have the functions derived from the equation \(4x^2 + 2y^2 = 1\), you can graph them to see the ellipse's shape.To graph the ellipse, follow these steps:
  • Use the equations \(y = \sqrt{\frac{1 - 4x^2}{2}}\) for the top half and \(y = -\sqrt{\frac{1 - 4x^2}{2}}\) for the bottom half.
  • Make a table of values for \(x\), calculating the corresponding \(y\) values for each equation.
  • Plot these points on a graph to visualize the halves forming a complete ellipse.
  • You can use graphing software or calculators to make this process easier and more accurate.
With the graph, it's helpful to understand how the ellipse aligns with the axes, and how its geometric properties relate back to the original equation.

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Most popular questions from this chapter

Factor the expression by grouping terms. $$x^{5}+x^{4}+x+1$$

(a) Find the radius of each circle in the pair, and the distance between their centers; then use this information to determine whether the circles intersect. (i) \((x-2)^{2}+(y-1)^{2}=9\) \((x-6)^{2}+(y-4)^{2}=16\) (ii) \(x^{2}+(y-2)^{2}=4\) \((x-5)^{2}+(y-14)^{2}=9\) (iii) \((x-3)^{2}+(y+1)^{2}=1\) \((x-2)^{2}+(y-2)^{2}=25\) (b) How can you tell, just by knowing the radii of two circles and the distance between their centers, whether the circles intersect? Write a short paragraph describing how you would decide this and draw graphs to illustrate your answer.

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