Teaching Notes 8.7: Describing Horizontal and Vertical Shifts
of the Graph of a Function
Just as with vertical shifts, equations describe the horizontal shifts of the graph of a function.
Students often confuse horizontal shifts with vertical shifts as the information of both can be
obtained from the numbers in the equation.- Discuss the graphs of the following functions: identity, squaring, cubing, square root, abso-
lute value, and reciprocal with your students. Also review the vertical shifts of these graphs.
If necessary, refer to 8.6: ‘‘Describing Vertical Shifts of the Graph of a Function.’’ - Explain that horizontal shifts are shifts tothe right or left on the coordinate plane. Note that
when a graph of a function is moved horizontally, the size and shape of the graph remains
the same. - Sketch the following examples of graphs of functions by plotting points:f(x)=x^2
andg(x)=(x−3)^2. Point out that the size and shape of both graphs are the same but
g(x)=(x−3)^2 is shifted 3 units to the right. Now sketch the graph ofh(x)=(x+1)^2 and
note that it has the same size and shape as the graph off(x)=x^2 butthatthegraphis
shifted 1 unit to the left. - Review the information and example on the worksheet with your students. For the example,
emphasize that students must identify the basic function and then find the values ofcand
d. Explain thatc=−4 because|x+ 4 |must be written as|x−(−4)|in the formula.d=− 2
so the graph is shifted down 2 units. Note thatctells the number of units the graph is shifted
horizontally.dindicates the number of units the graph is moved vertically. Also note that
the domains of the functions are restricted so that the functions are defined.
EXTRA HELP:
The number added to the value ofxindicates the number of units of a horizontal shift.
The number added to the function indicates the number of units of a vertical shift.ANSWER KEY:
The graph shows the following:(1)Squaring function shifted 1 unit to the right
andup2units.(2)Cubing function shifted 2 units to the right
and down 5 units.
(3)Reciprocal function shifted 3 units to the left. (4)Reciprocal function shifted 3 units up.
(5)Absolute value function shifted 1 unit to the
left and down 4 units.(6)Square root function shifted 2 units to the
left and down 6 units.
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(Challenge)Mikal is correct. The positive 2 that is added to the function indicates a shift up 2 units.
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