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Instructions and examples for graphing and transforming quadratic relations, including translations, reflections over axes, and symmetry. Students are asked to fill in tables and sketch graphs for various quadratic functions and discuss their properties.
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Name Math 1180 Due Monday November 24th Relations and Their Graphs
Circles.
1a. Let us first graph the relation x^2 + y^2 = 4. Do you know what to expect? Fill in the given table and graph the relation.
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b. Now we would like to graph the relation (x − 2)^2 + y^2 = 4. How is this relation similar to the previous one? In what way is it different? Fill in the table and graph
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c. Here we look at the relation (x − 2)^2 + (y + 3)^2 = 4. Again, how does this differ from the previous two relations? How is it similar? Fill in the table and graph.
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In the first problem, we started with a circle centered at the origin and in the next two steps, moved it around. This moving around is called “translating” the graph. We can translate other graphs too:
Parabolas.
2a. Make a table of at least three points, then sketch a graph of y = x^2.
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Symmetry and Reflections over axes.
Examine the following pictures. Notice how the coordinates change when we flip over the x axis, for instance. What happens when we flip over the y-axis? These pictures show “reflections” of a point over an axis. We can reflect more than just a point; we can reflect an entire graph.
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Figure 1
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Figure 2
3a. (Using Figure 2) What changes with respect to the coordinates when we reflect a single point over the y axis?
b. Now we would like to reflect the relation y = x^2 over the y axis. So we want to replace x with −x in our relation equation. Perform this replacement and simplify. What do you get?
c. When a relation has the same graph before and after reflecting over the y axis, we say that the relation is “symmetric” over the y axis. Is this relation symmetric with respect to the y axis?
4a. (Using Figure 1) Now look at the picture on the left above. What is different in the coordinates of a single point when we reflect it over the x axis?
b. If we want to reflect y = x^2 over the x axis, we need to replace y with −y in our equation. Do this and simplify.
c. Is this graph symmetric with respect to the x axis?
Symmetry and Reflections through/about the origin.
If we reflect a point across the x axis and then across the y axis as well, this is called “reflecting through (or about) the origin”.
5a. If we reflect over x and y axes, then algebraically, we replace x and y with their respective negatives to get a new relation. Algebraically reflect the unit circle through the origin (by replacing x with −x and y with −y) and give an equation for this new relation.
b. When the original relation is the same as the new one, we know that this relation is symmetric through the origin. Is the unit circle symmetric through the origin? Is it symmetric across the x axis? What about across the y axis?
6a. Use the method of replacements as in the previous problems to check symmetry to decide whether the relation 4y^2 + x^2 = 4 is symmetric. Is it symmetric across x? y? through the origin?
b. Using what you found in part a and the given portion of the graph below, graph the rest of the relation on the same graph.
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