Mapping Notation for
y=2log _3(-3x +9)-2
log_3 mean log base 3
not sure what that is, but it is the graph of
log_3(x)
reflected through the y-axis
shifted right by 3
stretched vertically by a factor of 2
shifted down by 2
Note that
log_3(-3x+9) = log_3(3) + log_3(3-x) = 1 + log_3(3-x)
This cancels the scale factor on the x-axis by shifting the graph up another unit in the y direction.
See
http://www.wolframalpha.com/input/?i=plot+y%3D2+log_3(-3x+%2B9)-2,+y%3Dlog_3(x),+y%3D1,+y%3D2+for+-3+%3Cx+%3C+6
To understand the mapping notation for the given equation, let's first analyze the equation itself.
The equation is y = 2log₃(-3x + 9) - 2.
Here, "log₃" represents the logarithm function with base 3. The expression inside the logarithm function is (-3x + 9).
The logarithm function with base 3, log₃, represents the power to which 3 must be raised to obtain the argument (-3x + 9).
In mapping notation, we typically express an equation by representing the input and output values as ordered pairs. For the given equation, we need to determine the range of values for x and the corresponding range of values for y. Here's how you can do that:
1. Start by selecting different values for x. These values should cover a range of possible inputs. Let's say we choose -1, 0, and 1.
2. Substitute these x-values into the original equation and solve for y. Using the values we selected, we get:
a) For x = -1: y = 2log₃(-3(-1) + 9) - 2
y = 2log₃(3) - 2 = 2(1) - 2 = 0
b) For x = 0: y = 2log₃(-3(0) + 9) - 2
y = 2log₃(9) - 2 = 2(2) - 2 = 2
c) For x = 1: y = 2log₃(-3(1) + 9) - 2
y = 2log₃(6) - 2 = 2log₃(2 · 3) - 2
y = 2(log₃(2) + log₃(3)) - 2 = 2(0.631 + 1) - 2 ≈ 1.262
3. Now, we have the corresponding y-values for the chosen x-values. We can represent this information in mapping notation as follows:
(-1, 0)
(0, 2)
(1, 1.262)
This mapping notation shows the pairs of input and output values for the equation y = 2log₃(-3x + 9) - 2.
Note that this is just one way to find the mapping notation for a given equation. Depending on the specific equation, you may need to select different values for x to get a more complete understanding of the mapping.