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 isâ€‹
(Notice that the domain of the square root function is all x â‰¥ 0 because you cannot take the square root of a negative number. The range is y â‰¥ 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 â‰¤ 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 â‰¤ 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 = â€“f(x). To reflect a function y = f(x) about the y-axis, multiply the interior of the function by negative one: y = f(â€“x). To summarize, we have
To reflect about the x-axis: |
y = â€“f(x) |
To reflect about the y-axis: |
y = f(â€“x) |
Reflections and translations can be combined. Letâ€™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 â‰¥ 0 and the range is now y â‰¤ 0.)
(Notice that the domain is now x â‰¤ 0 and the range is still y â‰¤ 0.)
(Notice that the domain is now x â‰¤ 2 and the range is still y â‰¤ 0.)
(Notice that the domain is still x â‰¤ 2 and the range is now y â‰¤ 1.)