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).
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 .
3 answers
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
. 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.
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.
1 answer