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

Question

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

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 2

which one is correct? a b c or d

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!