The function g(x) is a transformed image of the graph of f(x)=2^x. Write the equation of the transformed function g(x) in the form g(x)= b^x-h+k.
f(x)
(0,1)
(2,4)
g(x)
(-1,2)
(-3,-1)
2^(-1-h)+k = 2
2^(-3-h)+k = -1
solve for h and k
Of course many other values for b could have been chosen, but since only two data points were given, something had to be assumed.
just in case you are having trouble solving exponential equations,
eliminate k and you have
2 - 2^(-1-h) = -1 - 2^(-3-h)
3 = 2^(-1-h) - 2^(-3-h)
3 = 2^-h/2 - 2^-h/8
3 = 2^-h (1/2 - 1/8)
3 = 2^-h * 3/8
8 = 2^-h
h = -3
Then k = -2, and
g(x) = 2^(x+3)-2
See the graph at
http://www.wolframalpha.com/input/?i=2%5E(x%2B3)-2+for+-4+%3C+x+%3C+0
To determine the equation of the transformed function g(x), we can use the given coordinate points and analyze how they relate to the original function f(x)=2^x.
Step 1: Determine the vertical transformation (h)
The point (0,1) on f(x) corresponds to the point (-1,2) on g(x). Since the x-coordinate has changed from 0 to -1, there is a horizontal shift by 1 unit. Therefore, h = 1.
Step 2: Determine the vertical transformation (k)
The point (0,1) on f(x) corresponds to the point (-1,2) on g(x). Since the y-coordinate has changed from 1 to 2, there is a vertical shift by 1 unit. Therefore, k = 1.
Step 3: Determine the base of the exponential function (b)
Since the base of the original function f(x) is 2, we can assume that the base of the transformed function g(x) is also 2. Therefore, b = 2.
Step 4: Write the equation of the transformed function g(x)
Using the values determined in Steps 1-3, we can write the equation of the transformed function g(x) in the form g(x) = b^x - h + k.
g(x) = 2^x - 1 + 1
Simplifying the equation, we get:
g(x) = 2^x
Therefore, the equation of the transformed function g(x) is g(x) = 2^x.
To find the equation of the transformed function g(x) in the form g(x) = b^x - h + k, we need to determine the values of b, h, and k.
First, let's look at how the function f(x) = 2^x has been transformed to obtain g(x). We have two given points for f(x) and g(x):
For f(x):
(0, 1) and (2, 4)
For g(x):
(-1, 2) and (-3, -1)
To find the value of b, we need to identify how the exponential base, 2, has been transformed.
In f(x), when x increases by 1, the value of y (2^x) doubles. This indicates a vertical stretch by a factor of 2.
Since the given point (-1, 2) is on the graph of g(x), we can determine that g(-1) = 2. Plugging this into the equation, we have:
2 = b^(-1)
2 = 1/b
Solving this equation, we find that b = 1/2.
Now, let's find the values of h and k, which represent horizontal and vertical shifts, respectively.
Comparing the x-values of the given points for f(x) and g(x), we can determine the horizontal shift (h):
For f(x):
x = 0 and x = 2
For g(x):
x = -1 and x = -3
By comparing the corresponding x-values, we can observe that g(x) has been shifted 1 unit to the left. So, h = 1.
Next, we need to find the vertical shift (k). By comparing the y-values of the given points for f(x) and g(x), we can determine the shift in the y-direction:
For f(x):
y = 1 and y = 4
For g(x):
y = 2 and y = -1
We can observe that g(x) has been shifted 1 unit downward. So, k = -1.
Now, we have all the values we need to write the equation of the transformed function g(x) in the form g(x) = b^x - h + k:
g(x) = (1/2)^x - 1 + (-1)