Question Video: Identifying a Logarithmic Function with a Horizontal Translation from Its Graph Mathematics • 10th Grade

Which function represents the graph?

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Video Transcript

Which function represents the following graph?

We are given five logarithmic functions to consider. 𝑓 of 𝑥 equals log base three of 𝑥. 𝑓 of 𝑥 equals log base three of three 𝑥. 𝑓 of 𝑥 equals log base three of 𝑥 plus three. 𝑓 of 𝑥 equals log base 10 of 𝑥 plus three. And 𝑓 of 𝑥 equals log base three of negative 𝑥 plus three.

Our initial observation of the graph tells us that the shape resembles that of the graph of a logarithmic function. We can also see from the graph that this function has an 𝑥-intercept of negative two and a 𝑦-intercept of positive one. There is a vertical asymptote at 𝑥 equals negative three. No part of the graph is to the left of the asymptote, and the function is increasing.

By examining the options given, we are being asked to find which transformation of a logarithmic function fits the graph. If this function were of the form 𝑓 of 𝑥 equals log base 𝑛 of 𝑥, then the graph would have an 𝑥-intercept of one and pass through the point 𝑛, one.

A logarithmic function of this form will also have a vertical asymptote of 𝑥 equals zero. Since the function given in option (A) is of this form, with 𝑛 equal to three, we expect it to contain the points one, zero and three, one. If we plot these points on the coordinate plane, we see that they do not lie on the graph. But if we translate these points and the vertical asymptote three units to the left, they match the graph exactly. We therefore eliminate option (A).

It is our job now to determine which of the remaining options is a horizontal translation of this first function three units to the left. We can confidently eliminate option (D), since it is not base three. A logarithmic function with base 10 would be more flat compared to the log base three function.

So we have options (B), (C), and (E) left to consider. We notice that function (B) is of the form log base three of 𝑏 times 𝑥. This form represents a horizontal stretch by a scale factor of one over 𝑏. Therefore, option (B) represents a horizontal stretch by a scale factor of one-third. But we are looking for the function that horizontally translates the graph three to the left.

So we eliminate option (B) and move on to option (C), which is of the form log base three of 𝑥 plus ℎ. This type of transformation causes the log base three of 𝑥 function to translate or shift by negative ℎ horizontally. Since ℎ equals three, this transformation shifts the log base three function by negative three, in other words three to the left. This matches the horizontal translation shown in the diagram.

To be confident in our answer, it’s a good idea to check the function using the two points we found from the graph. We will evaluate 𝑓 of negative two and 𝑓 of zero to confirm the coordinate points negative two, zero and zero, one are given by log base three of 𝑥 plus three.

Evaluating 𝑓 of negative two in the function from option (C) gives us log base three of one. Using the knowledge that a logarithm is the inverse of an exponential, we’re able to show that log base three of one equals zero. This is because three to the power of zero equals one. We can also verify 𝑓 of zero equals one because three to the power of one equals three.

We will now finish this problem by checking these two points in the function provided by option (E). In this case, 𝑓 of negative two equals log base three of five, which is not equal to zero. We note that 𝑓 of zero does equal one, but we have enough evidence to eliminate option (E) because we showed that it does not contain the point negative two, zero.

In conclusion, the function which represents the logarithmic graph given is 𝑓 of 𝑥 equal to log base three of 𝑥 plus three.