the graph of y=-1/2 x+5 +3 can be obtained from the graph of y= x by which transformations
what is the horizontal shift
by what factor is the graph stretched or shrunk vertically and how is it reflected
Assuming you meant -1/2 (x+5) + 3,
starting with y = x
shrink by a factor of 2 to get
y = 1/2 x
reflect through x-axis to get
y = -1/2 x
shift left 5 units to get
y = -1/2 (x + 5)
shift up 3 units to get
y = -1/2 (x+5) + 3
To understand the transformation that is applied to the graph of y = x to obtain the graph of y = -1/2x + 5 + 3, let's break it down step by step.
1. Horizontal Shift:
The equation y = -1/2x + 5 + 3 is in the form y = f(x) + k, where f(x) represents a function. The value of k, which is 8 in this case (5 + 3), indicates a vertical shift. Therefore, there is no horizontal shift because the x term is not affected.
2. Vertical Stretch/Shrink and Reflection:
The coefficient of x, -1/2, determines how the graph is vertically stretched or shrunk and reflected.
- Vertical Stretch/Shrink:
The absolute value of the coefficient tells us the stretch or shrink factor. In this case, the absolute value of -1/2 is 1/2. It indicates that the graph of y = -1/2x is vertically shrunk by a factor of 1/2. The original graph is squeezed vertically.
- Reflection:
Since the coefficient of x, -1/2, is negative, it indicates a reflection across the x-axis. The graph of y = -1/2x is flipped vertically.
To summarize, the transformations applied to the graph of y = x to obtain the graph of y = -1/2x + 5 + 3 are:
- No horizontal shift
- Vertical shrink by a factor of 1/2
- Reflection across the x-axis.