Describe the transformation in each function as it relates to the graph of f(x)=x A. g(x)=(-1/2x) B. g(x)=(2x)+3

Answer 1

f(x) =x is a straight line with slope =1, the same as a 45 degree angle with the x axis. It's upward sloping

g(x) = (-1/2x) is downward sloping with a slope =-1/2.

both f(x) = x and g(x) =(-1/2x) have x and y intercepts at the origin (0,0).

that's assuming you really meant to write g(x) = (-1/2)x and not (-1/2x) which means -1/(2x)

if g(x) = -1/(2x), it's a rectangular hyperbola and no longer a straight line, but two curved branches with a discontinuous point at x=0

g(x) =2x+3 is a vertical shift of f(x) up by 3 units and a vertical stretch by a factor of 2, as the straight line now has slope =2.

Answer 2

f(x) = x

A. g(x) = (-1/2)x

The slope of the function changes from a slope of 1 in f(x) to a slope of -1/2 in g(x). On a graph, this means the line of the function changes from having a positive slope to a negative slope and decreases the magnitude of the slope by 1/2.

B. g(x) = 2x+3

The slope of the function changes from a slope of 1 in f(x) to a slope of 2 in g(x). The y-intercept also changes from 0 in f(x) to 3 in g(x). On a graph, this means the magnitude of the slope increases by 2, and the line of the function is translated up by 3 units.

Answer 3

Multipliers grouped with the "x" are stretches or compressions in the "x" direction (horizontally) except they are opposite the way they appear. Example 1: g(x) = (5x) means the function f(x) = x is compressed by a factor of 5 in the x-direction (all x-values get multiplied by 1/5) meaning the point (5, 5) that was on f(x) = x, will now be transformed into the point (1, 5) for g(x) = (5x). Example 2: g(x) = (1/5x) means the function f(x) = x is stretched by a factor of 5 in the x-direction (all x-values get multiplied by 5) meaning the point (5, 5) that was on f(x) = x, will now be transformed into the point (25, 5) for g(x) = (1/5x).

Negative signs grouped with the "x" will result in reflections across the y-axis. Combinations such as g(x) = (-5x) mean the function f(x) = x is reflected across the y-axis and stretched by a factor of 5 meaning the point (1, 1) that was on f(x) = x will be transformed into (-1, 5) for g(x) = (-5x)

Numbers that are added to a function such as g(x) = x + 2 mean that the function is translated up by 2 units meaning the point (3, 3) that was on f(x) = x will be transformed into (3, 5) for g(x) = x + 2. Combinations such as g(x) = (3x) + 5 mean that the function f(x) = x is compressed horizontally by a factor of 3 (all x-values are multiplied by 1/3) and translated up by 5 units meaning the point (3, 3) that was of f(x) = x is now transformed into (1, 8). Notice that the x-value was multiplied by 1/3 and the y-value had 5 added to it.

These rules should help you do your problems.