How do I graph the function: y=2xx2−1?
1 Answer
I like to identify the following things first, when asked to graph a rational function:
- y-intercept(s)
- x-intercept(s)
- vertical asymptote(s)
- horizontal asymptote(s)
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To identify the y-intercept(s), ask yourself "what is the value of y when x=0"?
y=2(0)(0)2−1=0−1=0
y-intercept: (0,0) -
To identify the x-intercept(s), ask yourself "what is the value of x when y=0"?
For this problem, since we've already identified that the graph goes through (0,0), we have both the x-int and y-int complete! But in case you didn't realize...
0=2xx2−1 means that the numerator of the fraction must = 0
0=2x
0=x
x-intercept: (0,0) -
To identify the vertical asymptotes, we first try and simplify the function as much as possible and then look at where it is undefined
y=2xx2−1
y=2x(x+1)(x−1)
Undefined when denominator = 0:(x+1)(x−1)=0
Vertical asymptotes:x=−1,x=1 -
To identify the horizontal asymptotes, we think of the limiting behavior (ie: what happens as x gets HUGE)
y=2xx2−1→y=hugeHUGER→0
Horizontal asymptote:y=0
Now you might pick a couple additional points to the left/right/between your horizontal asymptotes to get a sense of the graph shape.
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Pick a point to the left of the
x=−1 asymptote, ie:x=−2
y=2(−2)(−2)2−1=−44−1=−43 Point 1:(−2,−43) -
Pick a point between the two asymptotes We already have the point (0,0) from above. Point 2:
(0,0) -
Pick a point to the right of the
x=1 asymptote, ie:x=2
y=2(2)(2)2−1=44−1=43 Point 3:(2,−43)
Domain:
Range:
