Adrienne said that a vertical stretch by a factor of 4, a horizontal shift to the left 1 unit, and a vertical shift down 3 units of the graph of the parent function f(x)=ex produce the graph of the function g(x)=4ex−1+3 . Explain whether Adrienne is correct and, if not, determine the transformations that produce the graph of the function g(x)=4ex−1+3 from f(x)=ex .
Adrienne is correct.
To analyze the transformations, we compare the equations of the two functions:
f(x) = ex
g(x) = 4ex - 1 + 3
1) Vertical stretch by a factor of 4:
The coefficient 4 in front of ex in g(x) stretches the graph vertically by a factor of 4 compared to f(x). This means that each y-coordinate in g(x) will be 4 times the corresponding y-coordinate in f(x). Since 4 is greater than 1, this corresponds to a vertical stretch.
2) Horizontal shift to the left 1 unit:
The term -1 in g(x) shifts the graph horizontally to the left by 1 unit compared to f(x). This means that for each x-value in g(x), we need to evaluate f(x+1) to get the corresponding y-value in f(x). So this shift to the left is also correct.
3) Vertical shift down 3 units:
The term +3 in g(x) shifts the graph vertically downwards by 3 units compared to f(x). This means that for each y-value in g(x), we need to subtract 3 from the corresponding y-value in f(x). This vertical shift down is also correct.
Therefore, Adrienne is correct in stating that the given transformations produce the graph of g(x) from f(x).
Stefan said that the y-intercept for the graph of the function f(x)=3log(x−7)+1 is 1
. He also said that as x approaches positive infinity, the graph of f(x) approaches positive infinity. Describe which of Stefan's statements, if any, are incorrect, and, if so, correct the statements
Stefan's statement that the y-intercept for the graph of f(x) is 1 is incorrect.
To find the y-intercept, we set x = 0 and evaluate f(0):
f(0) = 3log(0 - 7) + 1
However, the expression inside the logarithm, (0 - 7), results in a negative value, which is not defined in the logarithm function. Therefore, we cannot evaluate f(0), and the y-intercept is undefined.
Now, let's analyze Stefan's statement about the behavior of the graph of f(x) as x approaches positive infinity.
As x approaches positive infinity, the term (x - 7) approaches infinity since subtracting a finite number (7) from a growing number (x) results in a larger positive value. Therefore, log(x - 7) approaches infinity.
However, multiplying this by 3 and then adding 1 does not necessarily mean that the entire expression 3log(x - 7) + 1 approaches positive infinity. It depends on the magnitude of the value of x as it approaches positive infinity. The overall behavior of the graph could be increasing but bounded, increasing without bounds, or even decreasing.
In summary, Stefan's statement about the y-intercept is incorrect. The y-intercept for the graph of f(x) is undefined, not 1. His statement about the behavior of the graph as x approaches positive infinity is not fully correct or incorrect since it depends on the specific values of x.