provide three characteristics of the function 𝒚=logx that remain unchanged under the following transformations: a vertical stretch by a factor of 3 and a horizontal compression by a factor of 3.
the new function is 3log(3x)
Unchanged are
the asymptote
the domain
the range
To determine the characteristics of the function 𝒚 = logx that remain unchanged under the given transformations (vertical stretch by a factor of 3 and horizontal compression by a factor of 3), let's first understand how these transformations affect the graph of the function.
1. Vertical Stretch by a factor of 3:
A vertical stretch by a factor of 3 means that the function's output values (y-values) will be stretched, making the graph taller. This is achieved by multiplying the function by a factor of 3. So, the transformed function is 𝒚 = 3logx.
2. Horizontal Compression by a factor of 3:
A horizontal compression by a factor of 3 means that the function's input values (x-values) will be compressed, making the graph narrower. This is achieved by dividing the input values (x-values) of the function by a factor of 3. So, the transformed function is 𝒚 = 3log(x/3).
Now, let's determine the characteristics of the function that remain unchanged under these transformations:
1. Domain:
The domain of a logarithmic function 𝒚 = logx is all positive real numbers (x > 0). Since both the vertical stretch and horizontal compression only affect the input values (x-values) and do not change the restrictions on the domain, the domain of the transformed function 𝒚 = 3log(x/3) remains the same as 𝒚 = logx: x > 0.
2. Asymptote:
The graph of the function 𝒚 = logx has a vertical asymptote at x = 0. Both the vertical stretch and horizontal compression do not affect the position of the asymptote. Therefore, the transformed function 𝒚 = 3log(x/3) will still have a vertical asymptote at x = 0.
3. Shape:
The shape of the graph of 𝒚 = logx is a gradually increasing curve that approaches the vertical asymptote (x = 0) but never crosses it. Both the vertical stretch and horizontal compression preserve the shape of the original function. Hence, the transformed function 𝒚 = 3log(x/3) will also have the same shape as 𝒚 = logx.
To summarize, the three characteristics of the function 𝒚 = logx that remain unchanged under a vertical stretch by a factor of 3 and a horizontal compression by a factor of 3 are:
1. Domain: x > 0
2. Vertical asymptote: x = 0
3. Shape: Gradually increasing curve approaching the vertical asymptote.