Describe the graph of y=1/2x-10 -3 compared to the graph of y=1/x? explain
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To describe the graph of y = 1/2x - 10 - 3 compared to the graph of y = 1/x, we need to understand the behavior and properties of each function.
Starting with y = 1/x, this is a rational function where x cannot be equal to zero since division by zero is undefined. As x approaches positive or negative infinity, y approaches zero. The graph of y = 1/x is a hyperbola that consists of two branches, one in Quadrant I and III, and the other in Quadrant II and IV.
On the other hand, y = 1/2x - 10 - 3 is a linear function with a slight modification from y = 1/2x. It is a straight line with a slope of 1/2 and a y-intercept of -3. As x increases, y increases, and as x decreases, y also decreases.
To compare the two graphs:
1. Intercept: The two graphs intersect when y = y. So, set 1/x equal to 1/2x - 10 - 3:
1/x = 1/2x - 13
Cross multiply and simplify:
2x = x(1/2x - 13)
2x = 1 - 13x
15x = 1
x = 1/15
Substitute x back into either equation to find y:
y = 1/x = 1/(1/15) = 15
Therefore, the two graphs intersect at coordinates (1/15, 15).
2. Slope: The slope of y = 1/2x is 1/2, while the slope of y = 1/2x - 10 - 3 remains the same. So, the two lines have the same slope.
3. Position: The graph of y = 1/2x - 10 - 3 is a translated version of y = 1/2x that is shifted downward by 13 units and left by 1/15 unit.
In summary, the graph of y = 1/2x - 10 - 3 intersects the graph of y = 1/x at (1/15, 15), has the same slope, and is a shifted version of the original function y = 1/2x.