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Problem 3 Medium Difficulty

The graph of $ y = f(x) $ is given. Match each equation with its graph and give reasons for your choices.

(a) $ y = f (x - 4) $
(b) $ y = f(x) + 3 $
(c) $ y = \frac{1}{3} f(x) $
(d) $ y = -f (x + 4) $
(e) $ y = 2f (x + 6) $

Answer

(a) (graph 3) The graph of $f$ is shifted 4 units to the right and has equation $y=f(x-4)$.
(b) (graph 1 ) The graph of $f$ is shifted 3 units upward and has equation $y=f(x)+3$.
(c) (graph 4 ) The graph of $f$ is shrunk vertically by a factor of 3 and has equation $y=\frac{1}{3} f(x)$.
(d) (graph 5) The graph of $f$ is shifted 4 units to the left and reflected about the $x$ -axis. Its equation is $y=-f(x+4)$.
(e) (graph 2) The graph of $f$ is shifted 6 units to the left and stretched vertically by a factor of 2. Its equation is $y=2 f(x+6)$.

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Video Transcript

Okay, here we see the graph of F, and what we see in this transformation is that it's going to shift right, four units. So the graph would roughly look like that. And when we look at our answer choices, we see that that is graph three for Part B. We have our original graph again, and now we look at y equals F of X Plus three, and that indicates a shift up three units. So that would look something like this. And when we look at our answer choices, that is, Answer one Okay for part C, we see that we're multiplying the function by 1/3. That's a vertical shrink by a factor of three, and that means it's going to be 1/3 is tall, so that would look roughly like that. And we look at her answer choices, and that is answer for okay for part deed. We see that we have a shift to the left four, and we also have a reflection across the X axis. So that's going to give us something that looks kind of like this, and we look at her answer choices, and that's answer five. And then finally for party. We see that we're going to have a shift to the left six and we're also going to have a vertical stretch by a factor of 22 times taller. So that's going to look roughly like this and we see that that is answer choice to.