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Reflections of Graphs

Many graphs can be obtained by reflecting a base graph by multiplying various places in the function by negative numbers. Take for example, the square root function . Its graph is

 
(Notice that the domain of the square root function is all x ≥ 0 because you cannot take the square root of a negative number. The range is y ≥ 0 because the graph touches the x-axis at the origin, is above the x-axis elsewhere, and increases indefinitely.)
 
To reflect this base graph about the x-axis, multiply the exterior of the square root symbol by negative one, :
 

(Notice that the range is now y ≤ 0 and the domain has not changed.)

 

To reflect the base graph about the y-axis, multiply the interior of the square root symbol by negative one, :

(Notice that the domain is now x ≤ 0 and the range has not changed.)
 
The pattern of the reflections above holds for all functions. So to reflect a function y = f(x) about the x-axis, multiply the exterior of the function by negative one: y = –f(x). To reflect a function y = f(x) about the y-axis, multiply the interior of the function by negative one: y = f(–x). To summarize, we have
 

To reflect about the x-axis:

y = –f(x)

To reflect about the y-axis:

y = f(–x)

 

Reflections and translations can be combined. Let’s reflect the base graph of the square root function  about the x-axis, the y-axis and then shift it to the right 2 units and finally up 1 unit:

 (Notice that the domain is still x ≥ 0 and the range is now y ≤ 0.)
 

(Notice that the domain is now x ≤ 0 and the range is still y ≤ 0.)

 

 (Notice that the domain is now x ≤ 2 and the range is still y ≤ 0.)
 

(Notice that the domain is still x ≤ 2 and the range is now y ≤ 1.)





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