Pre-Calc Reflecting Graphs: Symmetry 4.3 Using your grapher, sketch the following: 1.a. Graph y = x 2  a set of points we all (x,y) and y = - x 2  a.

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Pre-Calc Reflecting Graphs: Symmetry 4.3 Using your grapher, sketch the following: 1.a. Graph y = x 2  a set of points we all (x,y) and y = - x 2  a set of points we all (x,- y) Reflection in the x-axis ! b. Graph y = x 3 + 2x 2 again another set of points we call (x,y) and y = - (x 3 + 2x 2 ) also another set of points we call (x, - y) Reflection in the x-axis!

2.a. Graph y = x 2 – 1 - in blue a set of points we call (x,y) and y = | x 2 – 1 | -- in red this is a set of points we call (x, |y|) all y’s must be positive. All points below the x-axis, get reflected above the x-axis, and all points above the x-axis stay as is! b. Graph y = x(x-1)(x–3) in blue a set of points we call (x,y) and y = | x(x-1)(x-3) | in red again another set of points we call (x,|y|) once again all y’s are positive. All points below the x-axis, get reflected above the x-axis, and all points above the x-axis stay as is!

3.a. Graph y = 2x – 1  a set of points we call (x,y)  in blue and y = 2(-x) – 1  in red this is a set of points we call (-x,y) This causes a reflection about the y-axis All (x,y)  (- x, y) b. Graph y = (x) 1/2  in blue a set of points we call (x,y) and y = (-x) 1/2  in red a set of points we call (-x,y) Again this is a reflection about the y-axis All (x,y)  (- x, y)

4. a. Graph y = 2x + 1  in blue a set of points we call (x,y) and x = 2y + 1 in red a set of points we call (y,x) This is a reflection about the ‘identity’ axis y = x so all (x,y)  (y,x) b. Graph y = x 2  in blue a set of points we call (x,y) and x = y 2  a set of points we call (y,x) Once again  a reflection about the ‘identity’ axis y = x so all (x,y)  (y,x)

Reflections in the x-axis – A graph is a reflection in the x-axis if all (x,y) can be paired to (x,- y) Reflections in the y-axis – a graph is a reflection in the y-axis if all (x,y) can be paired to (- x, y) Reflections in the line y = x – A graph is a reflection in the line y = x if all (x,y) can be paired to (y,x) Reflections in the origin – A graph is a reflection in the origin if all (x,y) can be paired to (- x,- y) Example: Use symmetry to sketch the graph of: y 4 = x + 1 (Think: could you graph y = x 4 ?) So first trade places with ‘x’ and ‘y’ and then solve for ‘y’)

y 4 = x + 1  x 4 = y + 1, solve for y x 4 – 1 = y Graph and then let every (x,y) become (y, x) Line of symmetry a line that is the perpendicular bisector of any segment joining any pair of corresponding points Point of symmetry A point ‘0’ such that it is possible to pair the points of the graph in such a way that ‘0’ is the midpoint of the segment joining each pair. For quadratics: Axis of symmetry: x = - b 2a ( x, y ) For cubics Point of symmetry: x = - b  ( - b, f( - b ) 3a 3a 3a