To compare the graph of y = 2log3(x−4)+2 to the parent graph y = log3(x), we can examine the changes in the asymptotes, domain, and range.
Asymptotes:
The parent graph, y = log3(x), has vertical asymptote at x = 0. This means that as x approaches 0 from the left or right, the graph of y = log3(x) gets closer and closer to negative infinity.
For y = 2log3(x−4)+2, the asymptote follows the same vertical line, x = 0, since the 2log3(x) term does not affect the vertical shift. Therefore, the graph has a vertical asymptote at x = 0, just like the parent graph.
Domain:
The domain of the parent graph, y = log3(x), is all positive real numbers, x > 0, since the logarithm function is not defined for x ≤ 0.
For y = 2log3(x−4)+2, we need to consider the x value inside the logarithm. The x−4 inside the logarithm should be greater than 0 to avoid negative numbers inside the logarithm. Therefore, the x value must be greater than 4. Additionally, since the logarithm function is not defined for x ≤ 0, we also have to ensure that x−4 > 0, which implies x > 4.
Thus, the domain of y = 2log3(x−4)+2 is x > 4, where x is greater than 4.
Range:
The range of the parent graph, y = log3(x), is all real numbers, since the logarithm function can take on any real value.
For y = 2log3(x−4)+2, the 2log3(x−4) term determines the range of the function. The logarithm function is always negative for x > 1, so multiplying it by 2 makes it even more negative. Since 2log3(x−4) can be any negative value, adding 2 to it shifts the range upwards. Thus, the range of y = 2log3(x−4)+2 is y > 2, where y is greater than 2.
In summary, the graph of y = 2log3(x−4)+2 has a vertical asymptote at x = 0, a domain of x > 4, and a range of y > 2.
Describe how the graph of y=2log3(x−4)+2 compares to the parent graph. Discuss any changes in the asymptotes, domain, and range.
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