Problem 25 Sketch the graph of each ellipse... [FREE SOLUTION]

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Precalculus Functions and Graphs

Mark Dugopolski

$Math Studyset 91影视 Explanations$ Math

4 Edition

Chapter 10: Problem 25

Sketch the graph of each ellipse and identify the foci. $$9 x^{2}+y^{2}=9$$

Short Answer

Expert verified

The ellipse is centered at (0, 0) with vertices at (0, 卤3) and co-vertices at (卤1, 0). The foci are at (0, 卤2鈭�2).

Step by step solution

Write the equation in standard form

The given equation is $9x^2 + y^2 = 9$. To write it in standard form, divide the entire equation by 9:\[\frac{9x^2}{9} + \frac{y^2}{9} = \frac{9}{9}\]Simplify the equation:\[x^2 + \frac{y^2}{9} = 1\]

Identify the form of the ellipse and its parameters

The standard form of an ellipse is $\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1$. In this case:\[a^2 = 1\] and \[b^2 = 9\], so \[a = 1\] and \[b = 3\].For this ellipse, $a < b$, meaning it is a vertical ellipse.

Sketch the graph of the ellipse

To sketch the ellipse, plot the center at the origin (0, 0). Since $a = 1$, there are vertices at $ (0, \, \pm 3)$. Since $b = 3$, there are co-vertices at $ (\pm 1, \, 0) $. Draw the ellipse based on these points.

Calculate the foci

To find the foci, use the equation $c^2 = b^2 - a^2$. Here, $b = 3$ and $a = 1$:\[c^2 = 9 - 1 = 8\]\[c = \sqrt{8} = 2\sqrt{2}\]The foci of the ellipse are at $(0, \, \pm 2\sqrt{2})$.

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Key Concepts

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

ellipse equation

An ellipse is a special type of curve on a plane. It's defined by its equation. The general form of an ellipse equation is: \[Ax^2 + By^2 = C\] . Here, A, B, and C are constants. For our specific exercise, the given equation is \[9x^2 + y^2 = 9\] . Understanding this form is crucial for manipulating it to a more helpful form.

standard form of an ellipse

To make graphing easier, we convert any ellipse equation to its standard form. The standard form of an ellipse is: \[\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1\] . For our example, we start with: \[9x^2 + y^2 = 9\] . We then divide every term by 9. This gives us: \[\frac{9x^2}{9} + \frac{y^2}{9} = \frac{9}{9}\] . Simplify to get: \[x^2 + \frac{y^2}{9} = 1\] . Now, this matches the standard form \[\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1\] , with \[a^2 = 1\] and \[b^2 = 9\].

vertices and foci of an ellipse

Vertices and foci are the most critical points of an ellipse. For the equation in the form \[\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1\] , vertices occur at \[(0, \pm b)\] and co-vertices at \[(\pm a, 0)\]. Using our example, we find:

$a = 1$
$b = 3$

Thus, vertices are at \[(0, \pm 3)\] and co-vertices at \[(\pm 1, 0)\]. To find the foci, use: \[c^2 = b^2 - a^2\] with \[{b^2 = 9}\] and \[{a^2 = 1}\]. So, \[c = \sqrt{8} = 2\sqrt{2}\]. Thus, the foci are at \[(0, \pm 2\sqrt{2})\].

graphing ellipses

To graph an ellipse, follow these steps:

Identify the type: vertical or horizontal, based on values of $a$ and $b$.
Plot the center: the origin, if the equation is in its standard form.
Plot the vertices and co-vertices.
Draw the ellipse smoothly around these points.

In our example, since $a = 1$ is less than $b = 3$, it is a vertical ellipse. Center is at (0,0). Vertices are at (0,3) and (0,-3). Co-vertices at (1,0) and (-1,0). Foci at (0,\pm2\sqrt{2}). Connecting these properly will give the desired graph.

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91影视

Sketch the graph of each ellipse and identify the foci. $$9 x^{2}+y^{2}=9$$

Short Answer

Step by step solution

Write the equation in standard form

Identify the form of the ellipse and its parameters

Sketch the graph of the ellipse

Calculate the foci

Key Concepts

ellipse equation

standard form of an ellipse

vertices and foci of an ellipse

graphing ellipses

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