Answer: -2*g(x) + 2
This shifts the graph down 1 unit, reflects over the x axis, then vertically dilates by a factor of 2.
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Explanation:
Refer to the figures below. The screenshots may be a bit on the big side, so brainly might attempt to reduce the size. To fix this, click on the magnifying glass to get a better look. There are 4 images in total.
Figure 1 is the original graph your teacher gave you.
Figure 2 shows the translation of the parabola 1 unit down. This aligns the right piece of the red parabola to the green V curve. This junction location is (1,0).
Afterward, in figure 3, we reflect the parabola over the x axis to have the left endpoint of the parabola (-3,8) move to (-3,-8). Unfortunately the (-3,-8) doesn't match up with the right most piece of the blue cubic curve.
Luckily, a quick fix is to double the y coordinate from -8 to -16 and the two pieces match perfectly. The other junction is (-3,-16). This is shown in figure 4.
To summarize, we did these transformations in the order presented.
- Shift down 1 unit (figure 2)
- Reflect over the x axis (figure 3)
- Vertically stretch or dilate by a factor of 2 (figure 4)
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Now to shift down 1 unit, we subtract 1 from each g(x) output.
Therefore, we go from g(x) to g(x)-1
Reflect over the x axis to get -(g(x) - 1) = -g(x) + 1
Lastly, we double everything when applying the dilation mentioned to end up with the final answer: -2*g(x) + 2
You can use a tool like GeoGebra to check your work. It's the app I used to make the figures shown below.
