The function y = x becomes y= -2(3x+2)-5. Describe the order in which the transformations would take place.
1. The function is stretched vertically by a factor of 2
2. The function is compressed horizontally by a factor of 1/3
3. The graph is reflected into the x-axis
4. The graph translates 2 units to the left
5. The graph translates downwards 5 units
Is that correct?
The function is stretched vertically by a factor of 2
The function is compressed horizontally by a factor of 1/3
The graph is reflected into the x-axis
The graph translates 2 units to the left
The graph translates downwards 5 units
IS THAT CORRECT?
The function is compressed horizontally by a factor of 1/3 (or stretched vertically by 3)
y = 3 x
Translates 2 units up
y = 3x+2
reflects over x axis
y = -(3x+2)
stretch vertically by 2
y = -2(3x+2)
down 5
Yes, that is correct! To further understand the order of transformations, let's break down each step and see how the function y = x transforms into y = -2(3x + 2) - 5.
1. Vertical Stretch: The function is stretched vertically by a factor of 2. This means that every y-coordinate of the graph will be multiplied by 2, making it twice as tall.
2. Horizontal Compression: The function is compressed horizontally by a factor of 1/3. This means that the x-values of the graph will appear to be closer together. To achieve this, we divide the x-values by 1/3, which is equivalent to multiplying them by 3.
So far, the transformed function is y = x, stretched vertically by a factor of 2 and compressed horizontally by a factor of 1/3, resulting in y = (1/3)x.
3. Reflection in the x-axis: The graph is reflected into the x-axis. This means that every point on the graph will be flipped across the x-axis. We achieve this by multiplying the entire function by -1, which reflects the graph's y-values across the x-axis.
The transformed function now becomes y = -(1/3)x.
4. Horizontal Translation: The graph translates 2 units to the left. This means that every x-coordinate of the graph will shift 2 units to the left. To achieve this, we subtract 2 from each x-value.
The transformed function becomes y = -(1/3)(x + 2).
5. Vertical Translation: The graph translates downwards 5 units. This means that every y-coordinate of the graph will shift 5 units downwards. To achieve this, we subtract 5 from each y-value.
Finally, the transformed function is y = -(1/3)(x + 2) - 5, which matches the given equation y = -2(3x + 2) - 5.
So the correct order of transformations is: Vertical Stretch, Horizontal Compression, Reflection in the x-axis, Horizontal Translation, and Vertical Translation.