Question

The equation of this graph is y = 2/3x − 4 What transformation would need to happen for the equation of the line to be

Shift the graph up 4

Shift the graph left 4

Shift the graph down 3 and left 2

Shift the graph right 3, down 2

Answer 1

which one is correct? a b c or d

Answer 2

To determine how the graph of the line \( y = \frac{2}{3}x - 4 \) would be affected by the transformations stated, we can analyze each transformation step by step:

1. **Shift the graph up 4**:
- This would change the equation from \( y = \frac{2}{3}x - 4 \) to \( y = \frac{2}{3}x - 4 + 4 = \frac{2}{3}x \).

2. **Shift the graph left 4**:
- To shift the graph left 4, we replace \( x \) with \( x + 4 \), resulting in:
\[
y = \frac{2}{3}(x + 4) - 4 = \frac{2}{3}x + \frac{8}{3} - 4 = \frac{2}{3}x + \frac{8}{3} - \frac{12}{3} = \frac{2}{3}x - \frac{4}{3}.
\]

3. **Shift the graph down 3 and left 2**:
- For this transformation, we first shift left 2, which gives us:
\[
y = \frac{2}{3}(x + 2) - 4 = \frac{2}{3}x + \frac{4}{3} - 4 = \frac{2}{3}x + \frac{4}{3} - \frac{12}{3} = \frac{2}{3}x - \frac{8}{3}.
\]
- Next, we shift down 3:
\[
y = \frac{2}{3}x - \frac{8}{3} - 3 = \frac{2}{3}x - \frac{8}{3} - \frac{9}{3} = \frac{2}{3}x - \frac{17}{3}.
\]

4. **Shift the graph right 3, down 2**:
- To shift right 3, we replace \( x \) with \( x - 3 \), resulting in:
\[
y = \frac{2}{3}(x - 3) - 4 = \frac{2}{3}x - 2 - 4 = \frac{2}{3}x - 6.
\]
- Then we shift down 2:
\[
y = \frac{2}{3}x - 6 - 2 = \frac{2}{3}x - 8.
\]

In conclusion, the transformations would change the original equation \( y = \frac{2}{3}x - 4 \) as follows:

- **Shift the graph up 4**: \( y = \frac{2}{3}x \)
- **Shift the graph left 4**: \( y = \frac{2}{3}x - \frac{4}{3} \)
- **Shift the graph down 3 and left 2**: \( y = \frac{2}{3}x - \frac{17}{3} \)
- **Shift the graph right 3, down 2**: \( y = \frac{2}{3}x - 8 \)

Answer 3

Let's analyze the transformations you've provided with the original equation \( y = \frac{2}{3}x - 4 \) to identify which option has the required transformation.

1. **Shift the graph up 4** (Option A):
- New equation:
\[
y = \frac{2}{3}x - 4 + 4 = \frac{2}{3}x
\]

2. **Shift the graph left 4** (Option B):
- New equation:
\[
y = \frac{2}{3}(x + 4) - 4 = \frac{2}{3}x + \frac{8}{3} - 4 = \frac{2}{3}x - \frac{4}{3}
\]

3. **Shift the graph down 3 and left 2** (Option C):
- First shift left 2:
\[
y = \frac{2}{3}(x + 2) - 4 = \frac{2}{3}x + \frac{4}{3} - 4 = \frac{2}{3}x - \frac{8}{3}
\]
- Then shift down 3:
\[
y = \frac{2}{3}x - \frac{8}{3} - 3 = \frac{2}{3}x - \frac{8}{3} - \frac{9}{3} = \frac{2}{3}x - \frac{17}{3}
\]

4. **Shift the graph right 3, down 2** (Option D):
- First shift right 3:
\[
y = \frac{2}{3}(x - 3) - 4 = \frac{2}{3}x - 2 - 4 = \frac{2}{3}x - 6
\]
- Then shift down 2:
\[
y = \frac{2}{3}x - 6 - 2 = \frac{2}{3}x - 8
\]

Now, let's see which of these final equations matches our expected scenarios:

- Up 4: \( y = \frac{2}{3}x \)
- Left 4: \( y = \frac{2}{3}x - \frac{4}{3} \)
- Down 3 and left 2: \( y = \frac{2}{3}x - \frac{17}{3} \)
- Right 3, down 2: \( y = \frac{2}{3}x - 8 \)

Since you asked for the transformation corresponding to a specific result, but did not specify a particular end equation to match with, if you are asking which option corresponds to a substantial alteration, **shifting the graph right 3 and down 2 results finally in \( y = \frac{2}{3}x - 8 \)** (Option D), and that is often an impactful transformation.

If you have a specific target equation in mind for comparison, please provide it so I can help identify which transformation achieves that!