Sketch the graph of y=g(x) by starting with the graph of...

Sketch the graph of $y=g(x)$ by starting with the graph of $y=f(x)$ and using transformations. Track at least three points of your choice and the vertical asymptote through the transformations. State the domain and range of $g$. $f(x)=\log _{\frac{1}{3}}(x), g(x)=\log _{\frac{1}{3}}(x)+1$

Sketch the graph of $y=g(x)$ by starting with the graph of $y=f(x)$ and using transformations. Track at least three points of your choice and the vertical asymptote through the transformations. State the domain and range of $g$.
$f(x)=\log _{\frac{1}{3}}(x), g(x)=\log _{\frac{1}{3}}(x)+1$

Key Concepts

Logarithmic Functions

A logarithmic function is the inverse of an exponential function, characterized by a domain of positive real numbers and a range of all real numbers. It typically has a vertical asymptote at x = 0 and its graph increases or decreases depending on the base. Understanding logarithmic functions provides a basis for applying transformations to their graphs.

Vertical Shifts

A vertical shift is a type of transformation where a constant is added or subtracted from a function, moving the graph up or down without changing its shape. In the context of the given functions, adding 1 to the logarithmic function f(x) lifts the entire graph upward by one unit, which affects the y-coordinates of all points on the graph.

Graph Transformations (Tracking Points)

Tracking points during graph transformations involves monitoring key coordinates through the transformation process. By following specific points on the original graph and applying the transformation rules, one can precisely sketch the new graph and understand how the positions of these points change, which is essential for accurate graph construction.

Vertical Asymptote Transformation

The vertical asymptote of a function is a line where the function approaches infinity or negative infinity, indicating a boundary for the function's domain. When a vertical shift is applied to a logarithmic function, the asymptote remains unchanged because it is determined by the input variable. Recognizing this invariant property is crucial in sketching the graph after transformation.

Domain and Range Analysis

Analyzing the domain and range of a function involves determining the set of all possible input and output values. For logarithmic functions, the domain is typically restricted to positive values, while the range covers all real numbers. Applying vertical shifts can affect the output values (range) but not the domain, and a proper analysis of these characteristics is essential when describing the overall behavior of the transformed function.

Transcript

00:01 So we have our original function is log of x in base 1 third.

00:08 And our g of x function is just adding on 1 to that.

00:16 And so that is going to be a slide.

00:20 I'll put parentheses around that.

00:23 This is going to just make the graph go up one.

00:26 So let's get some basic points that we can graph here.

00:30 And we know we have to plug in numbers that are positive.

00:34 It's impossible to put a negative number in here.

00:36 But we want to put special numbers in so that we can figure them out.

00:39 And we know this is like one -third to some power has to equal this value that we put in, this x.

00:47 And again, this is our answer, our number.

00:52 We'll consider that our number.

00:54 And so we want to put the appropriate powers of one -third in there.

00:58 So if i plug in has to be something bigger than zero, and naturally if i plug in one -third, we're going to have that value will be at one.

01:12 What if i plug in, how about i plug in three? and if i plug in three, what power do we take one -third to end up getting three? we'll reciprocate it, so that becomes negative one.

01:27 And what if i want that value to end up being, let me slip this in here, one.

01:32 So if i, one third to what power will end up giving me one? well, that needs to be zero.

01:39 So if i grab those points, and we'll do that right here, well, let's get rid of that...

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