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Chapter 0: Problem 44

Sketch the graph of the first function by plotting points if necessary. Then use transformation(s) to obtain the graph of the second function. \(y=x^{2}, \quad y=(x-2)^{2}\)

Short Answer

Expert verified
To sketch the graph of the first function \(y=x^2\), we plotted a set of points and connected them with a smooth curve. We then identified a horizontal shift of 2 units to the right as the transformation needed to obtain the second function \(y=(x-2)^2\) from the first one. After applying this transformation, we plotted the new set of points for the second function and connected them with a smooth curve. Thus, the graph of the second function is the graph of the first function shifted 2 units to the right.

Step by step solution

01

Sketching the graph of the first function \(y=x^2\)

To plot the graph of the first function, we can select a few points for the variable x and then find the corresponding values of y. We can then plot these points on the x-y plane: For x = -3: \(y=(-3)^2=9\) For x = -2: \(y=(-2)^2=4\) For x = -1: \(y=(-1)^2=1\) For x = 0: \(y=(0)^2=0\) For x = 1: \(y=(1)^2=1\) For x = 2: \(y=(2)^2=4\) For x = 3: \(y=(3)^2=9\) Now that we have these points, we can plot them on the x-y plane, and connect them with a smooth curve to get the graph of the first function, \(y=x^2\).
02

Identifying the transformation

To find the transformation that we need to apply to the first function to obtain the second function, we will compare the two equations: \(y=x^2\) (first function) \(y=(x-2)^2\) (second function) We can see that the second function is obtained from the first function by replacing "x" with "(x-2)". This transformation causes a horizontal shift of the graph to the right by 2 units.
03

Applying the transformation to sketch the graph of the second function \(y=(x-2)^2\)

Now, let's apply the identified transformation (horizontal shift to the right by 2 units) to the graph of the first function. Take each point from the first function and shift it to the right by 2 units. We will get the following points: For x = -1: \(y=(-1-2)^2=9\) For x = 0: \(y=(0-2)^2=4\) For x = 1: \(y=(1-2)^2=1\) For x = 2: \(y=(2-2)^2=0\) For x = 3: \(y=(3-2)^2=1\) For x = 4: \(y=(4-2)^2=4\) For x = 5: \(y=(5-2)^2=9\) Now plot these points on the x-y plane and connect them with a smooth curve. This will be the graph of the second function, \(y=(x-2)^2\). By applying the transformation, we have sketched the graph of the second function from the graph of the first function.

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