Transformation and Function

When multiple translations are applied to a function, in what order are the translations done in?

SRT: stretches, reflections, transformations.

What is the inverse function of y=f(x)?

y=f(-1 an exponent)(x)

Invariant Point

Point on a graph that remains unchanged after a transformation is applied to it, any point on a curve that lies on a line of reflection.

Reflection

Transformation where each point of original graph has an image point resulting from a reflection in a line; may result in a change of orientation of a graph while preserving its shape.

Relating to stretches what would the equation y= af(x) do to a graph?

When y= af(x) is multiplied by a, the result is a vertical stretch about the x-axis by a factor of |a|. If a is < 0 the graph is also reflected in the x-axis.

Relating to stretches what would the equation y= f(bx) do to a graph?

When y= f(bx)is multiplied by b, a horizontal stretch occurs about the y-axis by a factor of 1/|b|. If b < 0 the graph is also reflected about the y-axis.

Relating to reflections what would the equations y= -f(x) and y= f(-x) do to a graph?

Y= -f(x) would be a vertical reflection over the x-axis and (x.y) would become (x, -y) Y= f(-x) would be a horizontal reflection over the y-axis and (x,y) would become (-x,y).

In the equaution y=f(x-h), what happens to a graph if h is positive vs. when h is negative?

If h is negative (y=f(x-4)) then a horizontal translation 4 units right occurs, if h is positive (y=f(x+4)) then a horizontal translation 4 units left occurs.

In the equation y-k=f(x), what happens to a graph if k is positive vs. when k is negative?

If k is positive (y=f(x)+4) then a vertical translation 4 units up occurs, if k is negative (y=f(x)-4) then a vertical translation 4 units down occurs.

What are the three ways to determine the inverse of a function?

Interchange x and y values: 1. (x,y) --> (y,x) 2. y= f(x) --> x= f(y) 3. Reflect in the line y=x

In the equation y= af[b(x-h)] + k what do all of the variables do in relation to a function?

In this equation, a: is a vertical stretch by factor a and if a < 0 a reflection about the x-axis also occurs, b is a horizontal stretch by the factor of 1/b and if b < 0 than a reflection about the y-axis occurs. H is a horizontal translation and k is a vertical translation.

Stretch

Transformation where the distance of each x or y coordinate from the line of reflection is multiplied by a scale factor; scale factors between 0 and 1 result in the point moving closer to the line of reflection and a scale factor greater than 1 results in the point moving farther away from the line of reflection.