Symmetry
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PowerPoint presentation that includes definition of symmetry (whether a curve is symmetric with respect to the x-axis, y-axis, and origin). Several examples included - some of which involve using algebra and others which use graphing to test for symmetry. Symmetric points are briefly described. Some real life examples included as well.
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Symmetry
- 1. Using Equations and Graphs to Test for Symmetry
- 2. What does this word mean? Can you think of some things in everyday life that are symmetrical?
- 4. A graph of an equation can be symmetric with respect to… The x-axis The y-axis The origin **You need to check for all three!**
- 5. A graph is said to be symmetric with respect to the x-axis if for every point (x, y) on the graph, the point (x, -y) is also on the graph.
- 6. A graph is said to be symmetric with respect to the y- axis if for every point (x, y) on the graph, the point (- x, y) is also on the graph.
- 7. A graph is said to be symmetric with respect to the origin if for every point (x, y) on the graph, the point (-x, -y) is also on the graph.
- 13. Plot the point (2, -4) and the point that is symmetric to it with respect to the a) x-axis b) y-axis c) origin
- 14. Plot the point (-3, -1) and the point that is symmetric to it with respect to the a) x-axis b) y-axis c) origin
- 15. A graph is said to be symmetric with respect to the x-axis if for every point (x, y) on the graph, the point (x, -y) is also on the graph. Method for Checking: Replace y by –y in the equation. If an equivalent equation results, the graph of the equation is symmetric with respect to the x-axis. A graph is said to be symmetric with respect to the y-axis if for every point (x, y) on the graph, the point (-x, y) is also on the graph. Method for Checking: Replace x by –x in the equation. If an equivalent equation results, the graph of the equation is symmetric with respect to the y-axis. A graph is said to be symmetric with respect to the origin if for every point (x, y) on the graph, the point (-x, -y) is also on the graph. Method for Checking: Replace x by –x and y by –y in the equation. If an equivalent equation results, the graph of the equation is symmetric with respect to the origin.
- 16. Test y = 1 𝑥 for symmetry to the x-axis, y-axis, and the origin. Find the intercepts.
- 17. Test y = 3𝑥 𝑥2+9 for symmetry to the x-axis, y- axis, and the origin. Find the intercepts.
- 18. Test 9x2 + 4y2 = 36 for symmetry to the x- axis, y-axis, and the origin. Find the intercepts.
- 19. Test y= x3 for symmetry to the x-axis, y-axis, and the origin. Find the intercepts.
Editor's Notes
- Definition from google: Symmetry is when one shape becomes exactly like another if you flip, slide or turn it. Reflections from one side to another.
- These are some examples found in real life How are they symmetric?
- There are other types of symmetry in math but we will only be concerned with these If someone asks: other types of symmetry include radial symmetry, point symmetry, diagonal symmetry, etc.
- It is a reflection over the x-axis The top half is reflected exactly over the x-axis – highlight x-axis in a different color Label some points and show how its opposite exists as well Remember art projects you did? Paint on one side and fold it in half and it would get on the other side? That’s what this is like, it’s like you’re folding your paper in half over the x-axis Demonstrate with hard copy of paper in class
- It is a reflection over the y-axis The top half is reflected exactly over the y-axis – highlight y-axis in a different color Label the points and show how it’s opposite exists on the dotted portion Just like the art projects but now you’re folding your paper in half over the y-axis Demonstrate with hard copy of paper in class
- Symmetry with respect to the origin is that same as a reflection about the y-axis followed by a reflection about the x-axis Every point AND its exact opposite needs to be included in the graph Pick some points and show how their exact opposites are included on this graph X and Y axis symmetry will be the easiest to see right away Origin symmetry might need a little testing, doesn’t always look like it right away – pick a couple of points and check
- First let’s think about x-axis symmetry… does this look like it fits the bill.. Can we fold the graph in half over the x-axis and get the same thing? Point (x, y) and (x, -y) need to be on the graph Let’s think about y-axis symmetry… Can we fold the graph in half over the y-axis and get the same thing? Point (x, y) and (-x, y) need to be on the graph What about origin symmetry? Point (x, y) and (-x, -y) need to be on the graph Let’s pick a couple of points to look at – good to pick a couple, not just 1 point Are their EXACT OPPOSITES also on this graph?
- This one is a little easier to tell just by looking at it What type of symmetry is displayed? The graph is symmetrical with respect to the y-axis
- This one is also easy to see This graph is symmetrical with respect to the x-axis
- What type of symmetry is displayed here?
- X-axis: we want to reflect this point over the x-axis, that means the y-value is going to become the opposite – it’s -4 so the opposite is +4 (2, 4) Y-axis: we want to reflect this point over the y-axis, that means the x value is going to become the opposite, it’s +2 so the opposite is -2 (-2, 4) Origin: We ant to reflect this point over the x-axis and the y-axis, that means the x value and the y value are both going to become the opposite, +2 becomes a -2 and -4 becomes a +4 (-2, 4)
- Try this one on your own! X-axis: (-3, 1) Y-axis: (3, -1) Origin: (3, 1)
- Underline the key terms/method Discuss verbally Explain that this looks like a lot but as we go through the problems it will make more sense and become easier, plus it all goes back to what we saw visually happening on our graphs earlier
- The main goal for using the equation method is to substitute in the correct values, and then try to manipulate the equation to see if you can get it back to the original form. If you can do this – the graph of the equation will demonstrate that type of symmetry First, let’s consider x-axis symmetry. We know that (x, y) and (x, -y) need to be on the graph for this to be true, so we plug in –y for y in the equation and see if we can finagle it to get it back to the original Now let’s test y-axis symmetry. We know that for this symmetry to occur the point (x, y) and (-x, y) must be on the graph. So, we plug in –x for x and once again see if we can manipulate the equation back into the original form Lastly, let’s check for origin symmetry. If a point (x, y) is on the graph, in order for it to be symmetric with respect to the origin, we need to have (-x, -y) also on the graph AKA THE EXACT OPPOSITE OF THE POINT. So, we will plug in –x for x and –y for y in the equation and once again see if we can manipulate it back into the original equation.
- Work through this problem, but ask the students questions to guide them through this problem together. The main goal for using the equation method is to substitute in the correct values, and then try to manipulate the equation to see if you can get it back to the original form. If you can do this – the graph of the equation will demonstrate that type of symmetry First, let’s consider x-axis symmetry. We know that (x, y) and (x, -y) need to be on the graph for this to be true, so we plug in –y for y in the equation and see if we can finagle it to get it back to the original – What do we need to do to test for x-axis symmetry? Now let’s test y-axis symmetry. We know that for this symmetry to occur the point (x, y) and (-x, y) must be on the graph. So, we plug in –x for x and once again see if we can manipulate the equation back into the original form – What do we need to do to test for y-axis symmetry? Lastly, let’s check for origin symmetry. If a point (x, y) is on the graph, in order for it to be symmetric with respect to the origin, we need to have (-x, -y) also on the graph AKA THE EXACT OPPOSITE OF THE POINT. So, we will plug in –x for x and –y for y in the equation and once again see if we can manipulate it back into the original equation. – What do we need to do to test for origin symmetry?
- Try this one on your own!