Describe how each graph is obtained from the graph of 𝔂 = ƒ(x...

Question

Describe how each graph is obtained from the graph of 𝔂 = ƒ(x).

d. 𝔂 = ƒ(2x + 1)

Accepted Answer

Hello there. Today we're going to solve the following practice problem together. So first off, let us read the problem and highlight all the key pieces of information that we need to use in order to solve this problem. Given the function Y equals G of X, which transformation describes the graph of Y equals G of 3 x + 2? Fantastic. So it appears for this particular problem we're asked to look at our original function, which is Y equals G of X, and we're trying to describe what happens exactly when the graph is transformed from Y equals GFX to Y equals G of 3 X + 2. What kind of transformation takes place, and that's our final answer that we're ultimately trying to solve for is how do we describe this transformation? So with that said, let's read off our multiple choice answers to see what our final answer might be. A is the graph of G of X is compressed horizontally by a factor of 3 and shifted 2/3 units to the right. B is the graph of GFX is stretched horizontally by a factor of 3, and shifted 2/3 units to the left. See the graph of G of X is stretched horizontally by a factor of 3 and shifted 2/3 units to the right. And D is the graphic G of X is compressed horizontally by a factor of 3 and shifted 2/3 units to the left. Fantastic. So, first off, in order to solve this problem, you need to note that since the input G is to be, 3 X plus 2, we can factor out a 3, and when we do that, we can say that G of 3 of X + 2/3. Fantastic. So now we can say that the expression inside follows the form G of C X plus H. And therefore, since we can now say that this our expression that we found when we factor out of 3 now matches this form, we can now identify our transformations. So we need to note that the term, and let's make a note of this, we can note that the term X + 2/3 follows the form G of X plus H. And we can say that according to our transformation rules, this G of X plus H represents a horizontal shift, and because it represents a horizontal shift, we could say that if H is greater than 0, that will mean that the graph of GFX will be shifted. 8 units to the left. So I'll use a capital L to denote left. So when H is greater than 0, then that means the graph of GFX is going to be shifted H units to the left. However, if H is less than 0, that will Mean that the graph of G of X, we could say that this graph of G of X is now going to be taking the absolute value of H, so meaning if H, even if it's a negative value, it's still going to be positive, so the absolute value of H units. To the right, which I'll use a capital R to denote right. So once again, if H is less than 0, then that means G of X's function will be graphed the absolute value of H units to the right. So, therefore, without a doubt, H in this case is going to be equal to 2/3. So that means that the graph of GFX is going to be shifted to the left by 2/3 units. Fantastic. So now, our next step is we need to note that we're going to let U equal X + 2/3. So that means that the function G of 3 multiplied by U will match the standard form G of C multiplied by U. So according to transformation rules, G of C multiplied by U will result in a horizontal compression by a factor of C, where it's important to note where C is greater than 1. So with that said, since we know that C is greater than 1, we can note that since in this case, C is equal to 3, the graph of G of X will be compressed horizontally by a factor of 3. So it's gonna be compressed, so I'll say CH, so compressed C. H compressed horizontally by a factor of 3. So compressed horizontally by a factor of 3, meaning it's going to become narrower. This graph's gonna narrow. So with that said, without a doubt, therefore, the graph of G of X is compressed horizontally by a factor of 3 and will be shifted 2/3 units to the left, and this corresponds with Multiple choice answer letter A, which once again is the graph of G of X is compressed horizontally by a factor of 3 and shifted 2/3 units to the right. Thank you so much for watching. Hopefully that helped, and I can't wait to see you in the next video. Bye.

Describe how each graph is obtained from the graph of 𝔂 = ƒ(x).

d. 𝔂 = ƒ(2x + 1)

Key Concepts

Function Transformation

Horizontal Shifts

Horizontal Scaling

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