desribe how the graph of each funtion compares to the graph of y = l x l (absolute value)
a) y = 1 2x l
b) y = l 1/2x l
c) 2y = l x l
d) 1/2y = l x l
i don't understand when it is vertical/horizontal and when it is expansion or compression
a is expanded, y is twice what it was before for any x.
d) is expanded, y is (2abs x ) twice what it was before.
a) is horizontal compression by 1/2
To understand how each function compares to the graph of y = |x| (absolute value), let's consider the effect each function has on the original graph.
a) y = 12x|l|
For y = 12x|l|, the graph is stretched vertically by a factor of 12. This means that every y-coordinate on the graph is multiplied by 12, resulting in a taller and narrower graph compared to y = |x|. It is still symmetric with respect to the y-axis and maintains the V-shape characteristic of the absolute value function.
b) y = |1/2x|
For y = |1/2x|, the graph is compressed horizontally by a factor of 2. This means that every x-coordinate on the graph is divided by 2, resulting in a wider and flatter graph compared to y = |x|. It is still symmetric with respect to the y-axis and maintains the V-shape characteristic.
c) 2y = |x|
For 2y = |x|, the graph remains the same as y = |x| since the equation only affects the y-values by a factor of 2. It is symmetric with respect to both x and y-axes and maintains the V-shape characteristic.
d) (1/2)y = |x|
For (1/2)y = |x|, the graph is compressed vertically by a factor of 2. This means that every y-coordinate on the graph is divided by 2, resulting in a shorter and wider graph compared to y = |x|. It is still symmetric with respect to both x and y-axes and maintains the V-shape characteristic.
To summarize:
- Vertical stretches/compressions scale the y-values.
- Horizontal stretches/compressions scale the x-values.
- The absolute value function is symmetric with respect to the y-axis and maintains a V-shape.