Thread: How to find intercepts, asymptotes, end behaviors, etc.

1) 2
x^2-2x-3
Find) Domain
Range
Is This Continuous?
Local min/max
Vertical Asymptote
Horizontal Asymptote
X Intercepts
Y Intercepts

for function



Domain
Domain is the set of values of x for which function is defined.
The given function is not defined at x = -1 and x = 3

So, domain

Range
Range is f(x) values (or y values) you get, when you put these domain x values in function f(x). Since the function has a horizontal Asymptote at y = 0, so, y = 0 is excluded from Range.

So, Range

Horizontal Asymptote (HA)

To find HA, divide Numerator and denominator of the function with highest degree (power) of x. Then take limit



divide with highest degree, ,





Now take



HA is y = 0

Vertical Asymptote (VA)

For VA, Denominator = 0

(x+1)(x-3) = 0

x = -1 and x = 3

The graph is discontinuous at x = -1 and x = 3

y-intercepts

For y-intercept, put, x = 0 in equation,

we get, y = -2/3

y-intercept = -2/3, the graph cuts y-axis at -2/3.

x-intercepts

For x-intercept, put, y = 0 in equation,



we do not get any answer from this, so there is no, x-intercept. i.e., the graph does not cuts x-axis

Please see attached graph, the green lines show the asymptotes.