Problem 23 - Algebra A Combined Function

Question

Graph each equation. $$ 4(x-1)^{2}+9(y+2)^{2}=36 $$

Step-by-Step Solution

Answer

The ellipse is centered at \((1, -2)\) with a semi-major axis of 3 and semi-minor axis of 2.

Step 1: Identify the type of conic section

The given equation \( 4(x-1)^{2}+9(y+2)^{2}=36 \) is in the standard form of an ellipse equation. The general form of an ellipse is \( \frac{(x-h)^2}{a^2} + \frac{(y-k)^2}{b^2} = 1 \) or \( \frac{(x-h)^2}{b^2} + \frac{(y-k)^2}{a^2} = 1 \). Since \( 4(x-1)^2 + 9(y+2)^2 = 36 \) can be rearranged into such forms, we recognize it as an equation of an ellipse.

Step 2: Simplify and rewrite the equation

To rewrite the equation in standard form, divide every term by 36: \[ \frac{4(x-1)^2}{36} + \frac{9(y+2)^2}{36} = \frac{36}{36} \]Simplifying each term, we get:\[ \frac{(x-1)^2}{9} + \frac{(y+2)^2}{4} = 1 \]

Step 3: Determine the center of the ellipse

In the equation \( \frac{(x-1)^2}{9} + \frac{(y+2)^2}{4} = 1 \), the center of the ellipse is given by the coordinates \((h, k)\). Here, \( h = 1 \) and \( k = -2 \). Thus, the center is at the point \((1, -2)\).

Step 4: Find the lengths of the semi-major and semi-minor axes

In the expression \( \frac{(x-1)^2}{9} + \frac{(y+2)^2}{4} = 1 \), \( a^2 = 9 \) and \( b^2 = 4 \). - The semi-major axis length \( a = \sqrt{9} = 3 \).- The semi-minor axis length \( b = \sqrt{4} = 2 \). Since \( a > b \), the ellipse is oriented horizontally.

Step 5: Sketch the ellipse

To graph the ellipse centered at \((1, -2)\):- From the center, move 3 units left and right along the x-axis, to \((-2, -2)\) and \((4, -2)\).- Move 2 units up and down along the y-axis, to \((1, 0)\) and \((1, -4)\).- Draw a smooth, oval shape through these points to complete the ellipse.

Key Concepts

Ellipse Equation

An ellipse is a type of conic section, which can be represented by its unique equation. The standard form of an ellipse equation is given by:

• Horizontal orientation: \( \frac{(x-h)^2}{a^2} + \frac{(y-k)^2}{b^2} = 1 \)
• Vertical orientation: \( \frac{(x-h)^2}{b^2} + \frac{(y-k)^2}{a^2} = 1 \)

Here, \((h, k)\) represents the center of the ellipse, while \(a\) and \(b\) represent the lengths of the semi-major and semi-minor axes. This type of equation allows us to easily identify the shape and positioning of the ellipse on the coordinate plane.
By rearranging and transforming general quadratic equations, we can convert them into this standard form to analyze their properties.

Conic Sections

Conic sections are the curves obtained by intersecting a cone with a plane. These include circles, ellipses, parabolas, and hyperbolas. Each shape has specific properties and equations that describe them. Ellipses, in particular, are closed curves that can look like stretched circles.
Understanding where an ellipse fits among the other conic sections is crucial as it helps identify its key characteristics, such as symmetry and curvature.

• Circles are special types of ellipses where the lengths of both axes are equal.
• Ellipses have two axes of symmetry: the major and minor axes.

These properties distinguish ellipses from other conic sections and help in understanding their graphical representation.

Graphing Ellipses

Graphing an ellipse involves translating the algebraic equation into a visual shape on a coordinate plane. To successfully graph an ellipse, there are a few steps to follow:

• Identify the center: Find \( (h, k) \) from the equation.
• Determine the orientation: Check if the ellipse is horizontal (if \(a > b\)) or vertical (if \(b > a\)).
• Calculate the lengths of the axes: Use the values of \(a\) and \(b\) from the standard form equation.

After finding these elements, you can plot the center point, move along the axes from the center distances defined by \(a\) and \(b\), and sketch the rounded shape connecting these extremities. This graphical method provides a clear visual understanding of the ellipse's dimensions and placement.

Center of Ellipse

The center of an ellipse is a pivotal point in understanding its geometry. It acts as the midpoint of the longest line (major axis) and the shortest line (minor axis) passing through the ellipse. In its standard form equation \(\frac{(x-h)^2}{a^2} + \frac{(y-k)^2}{b^2} = 1\), the center is represented by the coordinates \((h, k)\).This point helps to determine how the ellipse is positioned on the graph. Once the center is identified, it becomes easier to plot the ellipse by measuring the directions and lengths of the axes from this central location. Knowing the center also helps in determining symmetries and other underlying properties of the ellipse's shape.

Semi-Major and Semi-Minor Axes

The semi-major and semi-minor axes are key components in defining the shape of an ellipse. These axes give the ellipse its elongated form and are calculated from the values \(a\) and \(b\) in the standard form equation. The semi-major axis is the longer of the two, while the semi-minor axis is the shorter, distinguishing the ellipse's broader and narrower dimensions.Both axes originate from the center of the ellipse.

• Semi-major axis (\(a\)): Extends from the center to the farthest edge.
• Semi-minor axis (\(b\)): Extends from the center to the nearest edge.

These measurements not only shape the ellipse but also define its orientation on the graph. Understanding these two axes ensures a correctly drawn ellipse, reflecting its true proportions.