Difference between revisions of "005 Sample Final A, Question 16"
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Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle -x^2+4y^2-2x-16y+11=0}
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|Answer: | |Answer: | ||
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| − | |1) Since both x and y are squared it must be a hyperbola or an ellipse. We can conclude that the graph is | + | |1) Since both x and y are squared it must be a hyperbola or an ellipse. We can conclude that the graph is a hyperbola since <math>x^2</math> and <math>y^2</math> have the different signs, one negative and one positive. |
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| − | |2) Since | + | |2) Since the <math>y^2</math> is positive, the hyperbola opens up and down. |
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| − | |We start by | + | |We start by completing the square twice, once for x and once for y. After completing the squares we end up with <math>-(x + 1)^2 +4(y - 2)^2 = 4</math> |
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| + | |'''Common Mistake:''' When completing the square we will end up adding numbers inside of parenthesis. So make sure you add the correct value to this other side. In this case we add -1, and 16 for completing the square with respect to x and y, respectively. | ||
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Revision as of 10:27, 2 June 2015
Question Graph the following,
| Foundations: |
|---|
| 1) What type of function is this? |
| 2) What can you say about the orientation of the graph? |
| Answer: |
| 1) Since both x and y are squared it must be a hyperbola or an ellipse. We can conclude that the graph is a hyperbola since Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle x^2} and Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle y^2} have the different signs, one negative and one positive. |
| 2) Since the Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle y^2} is positive, the hyperbola opens up and down. |
Solution:
| Step 1: |
|---|
| We start by completing the square twice, once for x and once for y. After completing the squares we end up with Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle -(x + 1)^2 +4(y - 2)^2 = 4} |
| Common Mistake: When completing the square we will end up adding numbers inside of parenthesis. So make sure you add the correct value to this other side. In this case we add -1, and 16 for completing the square with respect to x and y, respectively. |
| Step 2: |
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| Now that we have the equation that looks like an ellipse, we can read off the center of the ellipse, (0, -1). |
| From the center mark the two points that are 3 units left, and three units right of the center. |
| Then mark the two points that are 2 units up, and two units down from the center. |
| Final Answer: |
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| The four vertices are: Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle (-3, -1), (3, -1), (0, 1) \text{ and } (0, -3)} |
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