Section 3.5 Transformations Vertical Shifts (up or down) Graph, given f(x) = x 2. Every point shifts up 2 squares. Every point shifts down 4 squares.

Section 3.5 Transformations Horizontal Shifts (left or right) Graph, given f(x) = x 2. Every point shifts left 2 squares. Every point shifts right 4 squares.

Section 3.5 Transformations Vertical Stretching and shrinking. Graph, given f(x) = x 2. Every y coordinate of each point is multiplied by 2. Every y coordinate of each point is multiplied by 1/2. (1, 2)(-1, 2) (-2, 8)(2, 8) (0, 0) (1, 1)(-1, 1) (-2, 4)(2, 4) (1, 1/2)(-1, 1/2) (-2, 2) (2, 2) (0, 0)

Section 3.5 Transformations Horizontal Stretching and shrinking. Graph, given f(x) = |x|. Every x coordinate of each point is divided by 2. Every x coordinate of each point is divided by 1/2. (1, 2)(-1, 2) (-2, 4)(2, 4) (0, 0) (3, 3)(-3, 3) (-2, 2)(2, 2) (6, 3) (-6, 3) (-4, 2) (4, 2) (0, 0) Everything is opposite of vertical! Remember we don’t divide by fractions, we multiply by the reciprocal! (4, 4)(-4, 4)

Section 3.5 Transformations Reflections, flipping over x-axis or y-axis. Graph, given. Every point will flip over the x-axis. Every point will flip over the y-axis.

Transformations have a specific order… The ORDER OF OPERATIONS! 1 st 2 nd 3 rd 4 th Outside the function… affects y-coordinates. 1. If a is negative, then flips over x-axis. 2. If | a | is > 1, then Vertical Stretch. If 0 < | a | < 1, then Vertical Shrink. Inside the function… affects x-coordinates. 1. If b is negative, then flips over y-axis. 2. If | b | is > 1, then Horizontal Shrink. If 0 < | b | < 1, then Horizontal Stretch. Inside the function… affects x-coord. Solve bx – c = 0. The answer for x will tell you which direction (sign) and how far (value). Outside the function… affects y-coord. Take the value of d for face value. + d goes up d units; – d goes down d units. EXAMPLE. 1. Flips over y-axis – x + 5 = = x Right 5 units. ( ) Flips over x-axis. 3 & 4 4. Vertical Stretch by Up 3 units.

Consider the function on the graph. 1. x – 2 = 0. x = +2 Right 2 units. Graph. 2. Up 3 units Graph.

Consider the function on the graph. 1. x – 2 = 0. x = +2 Right 2 units. 1. negative on the inside flips over the y-axis. Graph. 2. Up 3 units Graph. 2. – 5 on the outside shifts down 5 units

Consider the function on the graph. 1. Negative on the 3 will flip the graph over the x-axis and | -3 | = 3 and will cause a vertical stretch by multiplying the y-coordinates by 3. Graph. 1. A quicker way is to multiply all y-coordinates by -3. (3, -1) (1, -1) (-1, 1) (-3, 2) (-3, -6) (-1, -3) (1, 3) (3, 3)

Consider the function on the graph. 1. Negative on the 3 will flip the graph over the x-axis and | -3 | = 3 and will cause a vertical stretch by multiplying the y-coordinates by 3. Graph. 1. A quicker way is to multiply all y-coordinates by -3. (3, -1) (1, -1) (-1, 1) (-3, 2) (-3, -6) (-1, -3) (1, 3) (3, 3) 1. Multiplying ½ to the inside will cause a horizontal stretch. Remember, everything is opposite. Divide all x-coordinates by ½. Again we don’t divide by fractions, instead we multiply by the reciprocal ( times by 2). (-6, 2) (-2, 1) (2, -1) (6, -1)

Consider the function on the graph. 1. Negative on the 2 will flip the graph over the y-axis and | -2 | = 2 and will cause a horizontal shrink by dividing the x-coordinates by 2. Graph. 1. A quicker way is to divide all x-coordinates by -2. (3, -1) (1, -1) (-1, 1) (-3, 2)(1.5, 2) (0.5, 1) (-0.5, -1) (-1.5, -1)

Consider the function on the graph. Graph. (3, -1) (1, -1) (-1, 1) (-3, 2) (-6, 2) (-4, 1) (-2, -1) ( 0, -1) 1. The “+3” on the inside of the ( )’s will move every x-coordinate to the left 3 units. 2. The multiplication of 2 on the outside will multiply 2 to every y-coordinate. (-6, 4) ( 0, -2)(-2, -2) (-4, 2)