4.3 Transformations of Functions Reflecting Graphs; Symmetry -vertical shifts -horizontal shifts -reflections over x and y axis

Vertical Shifts Adding a constant to a function shifts the graph vertically. – UP if the constant is positive – DOWN if the constant is negative.

we will want to start with the parent function. – Parent functions are functions like. These are functions that we are familiar with, nothing has happened to them yet

y=x 2

y = x 2 y = x 2 + 2

y = x 2 y = x 2 - 3

Horizontal Shift To move the graph either left or right – LEFT you add a constant c to x – RIGHT you subtract a constant c to x

y = x 2 y = (x+3) 2

y = x 2 y = (x-2) 2 y = (x+4) 2

Consider any graph y = f(x) – We can now shift it up or down, and left and right one movement at a time, we cannot get off of the x or y axis. – But what if we want to move that graph y = f(x) up and left at the same time? (diagonally) So we can move the graph anywhere on the Cartesian Plane.

We can combine a vertical shift and a horizontal shift to move the graph of y = f(x) to any spot on the Cartesian Plane. Transformation Combinations

GraphSight Examples.

Vertical Stretching/Compressing y = f(x) Vertical stretching or compressing refers to how the y value changes between y = f(x) and y = af(x) at the same value x. Looking at y = f(x) and y = af(x), here the y value changes by a multiple of a.

Complete the following tables xy = x xy = 3x

y = x 2 y = 3x 2 Notice that the value that we multiplied f(x) by was 3. So each y value is now three times greater in y=af(x) than it was in y=f(x)

a > 1 When a > 1. Then the graph is stretched vertically 0 < a < 1 When 0 < a < 1. Then the graph is compressed vertically

Now “a” is between 0 and 1 so you can see that the pink graph has shrunk vertically.

Now when we stretch or shrink something horizontally, then we are multiplying each x value by a multiple a. Not the whole function, just the individual x values. Stretches when 0 < a < 1 Shrinks when a > 1

Horizontal Shrinking/Stretching

stretch shrink

Reflections Across the x axis (multiply the entire function by -1) Across the y axis (multiply each x by a -1)

Reflection in the y-axis The graph of y=f(-x) is obtained by reflecting the graph of y=f(x) in the y-axis. Notice that each point (x,y) on the original graph (red) becomes the point (-x,y) on the reflected blue graph.

Reflecting along the line y=x This refection requires the interchanging of x and y in the equation.

We’ve discussed how a quadratic has a vertex at –b/2a, it also has a line of symmetry through that point. A cubic has a point of symmetry at the point –b/3a.

Types of Symmetry X-AXIS  if (x,y) then (x,-y) Y-AXIS  if (x,y) then (-x,y) ORIGIN  if (x,y) then (-x,-y) Y=X  if (x,y) then (y,x)

X-AXIS  if (x,y) then (x,-y) (2,4) (2,-4)

Y-AXIS  if (x,y) then (-x,y) (2,4) (-2,4)

Some x-axis symmetry graphs

Y-axis symmetry graphs

Symmetry with respect to the origin Best way I can explain how to visually see if a graph is symmetric with respect to the origin is to fold the x-axis, and then fold the y- axis does the graph end up overlapping itself?

Symmetry about the line y=x. Notice that x and y values are interchanged.

To check algebraically if an equation is symmetric about any of these 3 lines and 1 point, then you simply apply the rule to each equation and see if the result is an equivalent equation to the original.

Find the symmetry for the equation x 2 + y 2 = 9. X-AXIS  if (x,y) then (x,-y) Y-AXIS  if (x,y) then (-x,y) ORIGIN  if (x,y) then (-x,-y) Y=X  if (x,y) then (y,x)

Find the symmetry for the equation x 4 = y - 3. X-AXIS  if (x,y) then (x,-y) Y-AXIS  if (x,y) then (-x,y) ORIGIN  if (x,y) then (-x,-y) Y=X  if (x,y) then (y,x)

Hwk. Pg , 7-12, 15, 16