MTH 161 Ch1 Transformations Symmetry and Extrema (1.6-1.7)

What are the transformation rules for f(x) + k and f(x) - k?

f(x) + k shifts the graph UP k units. f(x) - k shifts it DOWN k units. These changes are outside the function.

What do f(x + h) and f(x - h) do?

f(x + h) shifts LEFT h units. f(x - h) shifts RIGHT h units. Inside changes are opposite to intuition.

What does a outside do in a*f(x)?

If a is bigger than 1 it is a vertical stretch. If a is between 0 and 1 it is a vertical shrink. A negative a also reflects over the x-axis.

What does f(bx) do?

Horizontal stretch or shrink by a factor of 1/b. It acts inside the function, so it behaves opposite to what you expect.

How do you reflect over the x-axis and y-axis?

x-axis: multiply the outside by -1, giving -f(x). y-axis: replace x with -x, giving f(-x).

What order do you apply transformations?

Horizontal (left and right, inside) first, then vertical (outside).

How do you move a point through transformations?

Apply each transformation to the coordinates in order. A point (x, y) under f(x - 2) + 3 moves to (x + 2, y + 3).

How do you test for an even function?

Replace x with -x. If you get f(x) back, it is even. Even functions have only even powers and are symmetric to the y-axis.

How do you test for an odd function?

Replace x with -x. If you get -f(x), it is odd. Odd functions have only odd powers and are symmetric to the origin.

How do shifts affect symmetry?

Shifting up or down does not change y-axis symmetry. Shifting left or right breaks it.

How do you test x-axis symmetry?

Replace y with -y. If the equation is equivalent it has x-axis symmetry. Example: the circle x^2 + y^2 = 9.

What do increasing, decreasing and constant mean?

Increasing: y rises as x increases (uphill). Decreasing: y falls (downhill). Constant: y stays the same.

How do you write increasing and decreasing intervals?

Use x-values only, always with open parentheses.

What is a relative maximum and minimum?

Relative minimum: the function changes from decreasing to increasing. Relative maximum: it changes from increasing to decreasing.

What is an absolute max or min?

The single highest or lowest point of the entire function, such as the vertex of a parabola.

What is a parent functions reference sheet for?

It lists domain, range, intercepts and odd or even classification for the standard functions. Know the basic shapes before applying transformations.