Lesson 3-1 Symmetry and Coordinate Graphs. Symmetry with respect to the origin Two Steps: 1.Find...
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Transcript of Lesson 3-1 Symmetry and Coordinate Graphs. Symmetry with respect to the origin Two Steps: 1.Find...
![Page 1: Lesson 3-1 Symmetry and Coordinate Graphs. Symmetry with respect to the origin Two Steps: 1.Find f(-x) and –f(x) 2.If f(-x)=-f(x), the graph is symmetric.](https://reader037.fdocuments.in/reader037/viewer/2022102809/56649eff5503460f94c1560c/html5/thumbnails/1.jpg)
Lesson 3-1Symmetry and Coordinate
Graphs
![Page 2: Lesson 3-1 Symmetry and Coordinate Graphs. Symmetry with respect to the origin Two Steps: 1.Find f(-x) and –f(x) 2.If f(-x)=-f(x), the graph is symmetric.](https://reader037.fdocuments.in/reader037/viewer/2022102809/56649eff5503460f94c1560c/html5/thumbnails/2.jpg)
Symmetry with respect to the origin
Two Steps:1. Find f(-x) and –f(x)2. If f(-x)=-f(x), the graph is symmetric
with respect to the origin.
![Page 3: Lesson 3-1 Symmetry and Coordinate Graphs. Symmetry with respect to the origin Two Steps: 1.Find f(-x) and –f(x) 2.If f(-x)=-f(x), the graph is symmetric.](https://reader037.fdocuments.in/reader037/viewer/2022102809/56649eff5503460f94c1560c/html5/thumbnails/3.jpg)
Symmetry with respect to the x-axis, y-axis, the line y=x, and the line y=-x.
1. Substitute (a,b) into the equation.2. x-axis, substitute (a,-b)3. y-axis, substitute (-a,b)4. y=x, substitute (b,a)5. y=-x, substitute (-b,-a)6. Check to see which test produces equivalent
equations.
![Page 4: Lesson 3-1 Symmetry and Coordinate Graphs. Symmetry with respect to the origin Two Steps: 1.Find f(-x) and –f(x) 2.If f(-x)=-f(x), the graph is symmetric.](https://reader037.fdocuments.in/reader037/viewer/2022102809/56649eff5503460f94c1560c/html5/thumbnails/4.jpg)
Vocabulary
• Image Point – When applying point symmetry to a set of points, each point P in the set must have an image point P′ which is also in the set.
• Point Symmetry - Two distinct points P and P ′are symmetric with respect to a point M if and only if M is the midpoint of segment PP . ′Point M is symmetric with respect to itself.
![Page 5: Lesson 3-1 Symmetry and Coordinate Graphs. Symmetry with respect to the origin Two Steps: 1.Find f(-x) and –f(x) 2.If f(-x)=-f(x), the graph is symmetric.](https://reader037.fdocuments.in/reader037/viewer/2022102809/56649eff5503460f94c1560c/html5/thumbnails/5.jpg)
• Line Symmetry – Two distinct points P and P ′are symmetric with respect to a line l if and only if l is the perpendicular bisector of segment PP . A point P is symmetric to itself ′with respect to line l if and only if P is on l.
![Page 6: Lesson 3-1 Symmetry and Coordinate Graphs. Symmetry with respect to the origin Two Steps: 1.Find f(-x) and –f(x) 2.If f(-x)=-f(x), the graph is symmetric.](https://reader037.fdocuments.in/reader037/viewer/2022102809/56649eff5503460f94c1560c/html5/thumbnails/6.jpg)
![Page 7: Lesson 3-1 Symmetry and Coordinate Graphs. Symmetry with respect to the origin Two Steps: 1.Find f(-x) and –f(x) 2.If f(-x)=-f(x), the graph is symmetric.](https://reader037.fdocuments.in/reader037/viewer/2022102809/56649eff5503460f94c1560c/html5/thumbnails/7.jpg)
A figure that is symmetric with respect to a given point can be rotated 180° about that point and appear
unchanged.
![Page 8: Lesson 3-1 Symmetry and Coordinate Graphs. Symmetry with respect to the origin Two Steps: 1.Find f(-x) and –f(x) 2.If f(-x)=-f(x), the graph is symmetric.](https://reader037.fdocuments.in/reader037/viewer/2022102809/56649eff5503460f94c1560c/html5/thumbnails/8.jpg)
The origin is a common point of symmetry.
The values in the tables suggest that f(-x)=-f(x) whenever the graph of a function is symmetric with respect to the origin.
![Page 9: Lesson 3-1 Symmetry and Coordinate Graphs. Symmetry with respect to the origin Two Steps: 1.Find f(-x) and –f(x) 2.If f(-x)=-f(x), the graph is symmetric.](https://reader037.fdocuments.in/reader037/viewer/2022102809/56649eff5503460f94c1560c/html5/thumbnails/9.jpg)
Determine whether the graph is symmetric with respect to the origin.
![Page 10: Lesson 3-1 Symmetry and Coordinate Graphs. Symmetry with respect to the origin Two Steps: 1.Find f(-x) and –f(x) 2.If f(-x)=-f(x), the graph is symmetric.](https://reader037.fdocuments.in/reader037/viewer/2022102809/56649eff5503460f94c1560c/html5/thumbnails/10.jpg)
We can verify by following two steps:
![Page 11: Lesson 3-1 Symmetry and Coordinate Graphs. Symmetry with respect to the origin Two Steps: 1.Find f(-x) and –f(x) 2.If f(-x)=-f(x), the graph is symmetric.](https://reader037.fdocuments.in/reader037/viewer/2022102809/56649eff5503460f94c1560c/html5/thumbnails/11.jpg)
Symmetric to Origin
![Page 12: Lesson 3-1 Symmetry and Coordinate Graphs. Symmetry with respect to the origin Two Steps: 1.Find f(-x) and –f(x) 2.If f(-x)=-f(x), the graph is symmetric.](https://reader037.fdocuments.in/reader037/viewer/2022102809/56649eff5503460f94c1560c/html5/thumbnails/12.jpg)
Determine whether the graph is symmetric with respect to the origin.
The graph appears to be symmetric with respect to the origin.
1. Find f(-x) and - f(x).2. If f(-x) = - f(x), the graph has point symmetry.
![Page 13: Lesson 3-1 Symmetry and Coordinate Graphs. Symmetry with respect to the origin Two Steps: 1.Find f(-x) and –f(x) 2.If f(-x)=-f(x), the graph is symmetric.](https://reader037.fdocuments.in/reader037/viewer/2022102809/56649eff5503460f94c1560c/html5/thumbnails/13.jpg)
![Page 14: Lesson 3-1 Symmetry and Coordinate Graphs. Symmetry with respect to the origin Two Steps: 1.Find f(-x) and –f(x) 2.If f(-x)=-f(x), the graph is symmetric.](https://reader037.fdocuments.in/reader037/viewer/2022102809/56649eff5503460f94c1560c/html5/thumbnails/14.jpg)
Determine whether the graph is symmetric with respect to the origin.
The graph is not symmetric with respect to the origin.
![Page 15: Lesson 3-1 Symmetry and Coordinate Graphs. Symmetry with respect to the origin Two Steps: 1.Find f(-x) and –f(x) 2.If f(-x)=-f(x), the graph is symmetric.](https://reader037.fdocuments.in/reader037/viewer/2022102809/56649eff5503460f94c1560c/html5/thumbnails/15.jpg)
• Line Symmetry – Two distinct points P and P ′are symmetric with respect to a line l if and only if l is the perpendicular bisector of segment PP . A point P is symmetric to itself ′with respect to line l if and only if P is on l.
Graphs that have line symmetry can be folded along the line of symmetry so that the two halves match exactly. Some graphs have more than one line of symmetry.
![Page 16: Lesson 3-1 Symmetry and Coordinate Graphs. Symmetry with respect to the origin Two Steps: 1.Find f(-x) and –f(x) 2.If f(-x)=-f(x), the graph is symmetric.](https://reader037.fdocuments.in/reader037/viewer/2022102809/56649eff5503460f94c1560c/html5/thumbnails/16.jpg)
Line Symmetry
![Page 17: Lesson 3-1 Symmetry and Coordinate Graphs. Symmetry with respect to the origin Two Steps: 1.Find f(-x) and –f(x) 2.If f(-x)=-f(x), the graph is symmetric.](https://reader037.fdocuments.in/reader037/viewer/2022102809/56649eff5503460f94c1560c/html5/thumbnails/17.jpg)
Line Symmetry
![Page 18: Lesson 3-1 Symmetry and Coordinate Graphs. Symmetry with respect to the origin Two Steps: 1.Find f(-x) and –f(x) 2.If f(-x)=-f(x), the graph is symmetric.](https://reader037.fdocuments.in/reader037/viewer/2022102809/56649eff5503460f94c1560c/html5/thumbnails/18.jpg)
Some common lines of symmetry are the x-axis, the y-axis, the line y=x , and the line y=-x.
![Page 19: Lesson 3-1 Symmetry and Coordinate Graphs. Symmetry with respect to the origin Two Steps: 1.Find f(-x) and –f(x) 2.If f(-x)=-f(x), the graph is symmetric.](https://reader037.fdocuments.in/reader037/viewer/2022102809/56649eff5503460f94c1560c/html5/thumbnails/19.jpg)
![Page 20: Lesson 3-1 Symmetry and Coordinate Graphs. Symmetry with respect to the origin Two Steps: 1.Find f(-x) and –f(x) 2.If f(-x)=-f(x), the graph is symmetric.](https://reader037.fdocuments.in/reader037/viewer/2022102809/56649eff5503460f94c1560c/html5/thumbnails/20.jpg)
![Page 21: Lesson 3-1 Symmetry and Coordinate Graphs. Symmetry with respect to the origin Two Steps: 1.Find f(-x) and –f(x) 2.If f(-x)=-f(x), the graph is symmetric.](https://reader037.fdocuments.in/reader037/viewer/2022102809/56649eff5503460f94c1560c/html5/thumbnails/21.jpg)
Determine whether the graph of x²+y= 3 is symmetric with respect to the x-axis, y-axis, the line y = x, the line y = -x, or none of these.
Substituting (a,b) into the equation yields a²+b=3 x axis (a,-b) a²-b=3y axis (-a,b) (-a)²+b=3 a²+b=3 y=x (b,a) b²+a=3y=-x (-b,-a) (-b)²+(-a)=3 b²-a=3
The graph is symmetric to the y axis
![Page 22: Lesson 3-1 Symmetry and Coordinate Graphs. Symmetry with respect to the origin Two Steps: 1.Find f(-x) and –f(x) 2.If f(-x)=-f(x), the graph is symmetric.](https://reader037.fdocuments.in/reader037/viewer/2022102809/56649eff5503460f94c1560c/html5/thumbnails/22.jpg)
The graph is symmetric to both the x and y axis.
![Page 23: Lesson 3-1 Symmetry and Coordinate Graphs. Symmetry with respect to the origin Two Steps: 1.Find f(-x) and –f(x) 2.If f(-x)=-f(x), the graph is symmetric.](https://reader037.fdocuments.in/reader037/viewer/2022102809/56649eff5503460f94c1560c/html5/thumbnails/23.jpg)
Classwork
![Page 24: Lesson 3-1 Symmetry and Coordinate Graphs. Symmetry with respect to the origin Two Steps: 1.Find f(-x) and –f(x) 2.If f(-x)=-f(x), the graph is symmetric.](https://reader037.fdocuments.in/reader037/viewer/2022102809/56649eff5503460f94c1560c/html5/thumbnails/24.jpg)