If I where to create a graph with an angled ellipse with given points of maximums, minimums, and right and left bounds, how would one derive the formula for the function assuming no outside technology can be used? Justify your answer by solving the following example: Lets say a maximum at (1.7,2.7), a minimum at (-1.4 ,-1), x-intercepts at x = -1.9 and 1.3, y-intercepts at 1.8 and -0.6, a left bound at (-2,-0.5), and a right bound at (2.8,1.8). 


To create a formula for an ellipse that is tilted or angled based on the given maximum, minimum, and boundary points, we can start by understanding that an ellipse in its general form (without rotation) has the equation:

\[
\frac{(x - h)^2}{a^2} + \frac{(y - k)^2}{b^2} = 1
\]

where \((h, k)\) is the center, \(a\) and \(b\) are the semi-major and semi-minor axes, and they represent the maximum distances along the \(x\)- and \(y\)-axes, respectively. When the ellipse is rotated by an angle \(\theta\), the equation becomes more complex:

\[
\frac{(x' - h')^2}{a^2} + \frac{(y' - k')^2}{b^2} = 1
\]

where \((x', y')\) are transformed coordinates under a rotation by \(\theta\), given by:

\[
x' = x \cos \theta + y \sin \theta, \quad y' = -x \sin \theta + y \cos \theta
\]

### Step-by-Step Solution

#### Step 1: Find the Center of the Ellipse
Since we are given points of maximum and minimum values, we can approximate the center of the ellipse by finding the midpoint between the maximum and minimum points.

Given maximum at \((1.7, 2.7)\) and minimum at \((-1.4, -1)\), the center \((h, k)\) of the ellipse is:

\[
h = \frac{1.7 + (-1.4)}{2} = 0.15, \quad k = \frac{2.7 + (-1)}{2} = 0.85
\]

So, the approximate center of the ellipse is \((0.15, 0.85)\).

#### Step 2: Calculate Semi-Major and Semi-Minor Axes Lengths
The length of the semi-major and semi-minor axes can be estimated from the distances between the maximum and minimum points along both axes. The maximum and minimum points indicate that the ellipse is elongated along a particular angle.

Using the distance formula, calculate the approximate length of the major axis between the maximum and minimum points:

\[
\text{Distance}_{\text{major axis}} = \sqrt{(1.7 - (-1.4))^2 + (2.7 - (-1))^2} = \sqrt{(3.1)^2 + (3.7)^2} \approx 4.82
\]

Since this is the full length of the major axis, the semi-major axis \(a\) is:

\[
a = \frac{4.82}{2} \approx 2.41
\]

For the semi-minor axis, consider the horizontal width, given the x-intercepts at \(x = -1.9\) and \(x = 1.3\):

\[
\text{Width}_{\

If I were to create a graph with an angled ellipse with given points of maximums, minimums, and right and left bounds, how would one derive the formula for the function assuming no outside technology can be used? Justify your answer by solving the following example: Let's say a maximum at (1.7,2.7), a minimum at (-1.4,-1), x-intercepts at x = -1.9 and 1.3, y-intercepts at 1.8 and -0.6, a left bound at (-2,-0.5), and a right bound at (2.8,1.8).

Learn how to derive the formula for an angled ellipse graph given maximum, minimum, x- and y-intercepts, and boundary points. This step-by-step solution covers key concepts in coordinate geometry, ellipse properties, and trigonometric transformations, tailored for advanced high school students.

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