Answer:
The first transformation occurs when we add a constant d to the parent function \displaystyle f\left(x\right)={b}^{x}f(x)=b
x
, giving us a vertical shift d units in the same direction as the sign. For example, if we begin by graphing a parent function, \displaystyle f\left(x\right)={2}^{x}f(x)=2
x
, we can then graph two vertical shifts alongside it, using \displaystyle d=3d=3: the upward shift, \displaystyle g\left(x\right)={2}^{x}+3g(x)=2
x
+3 and the downward shift, \displaystyle h\left(x\right)={2}^{x}-3h(x)=2
x
−3. Both vertical shifts are shown in Figure 5.
Graph of three functions, g(x) = 2^x+3 in blue with an asymptote at y=3, f(x) = 2^x in orange with an asymptote at y=0, and h(x)=2^x-3 with an asymptote at y=-3. Note that each functions’ transformations are described in the text.
Figure 5
Observe the results of shifting \displaystyle f\left(x\right)={2}^{x}f(x)=2
x
vertically:
The domain, \displaystyle \left(-\infty ,\infty \right)(−∞,∞) remains unchanged.
When the function is shifted up 3 units to \displaystyle g\left(x\right)={2}^{x}+3g(x)=2
x
+3:
The y-intercept shifts up 3 units to \displaystyle \left(0,4\right)(0,4).
The asymptote shifts up 3 units to \displaystyle y=3y=3.
The range becomes \displaystyle \left(3,\infty \right)(3,∞).
When the function is shifted down 3 units to \displaystyle h\left(x\right)={2}^{x}-3h(x)=2
x
−3:
The y-intercept shifts down 3 units to \displaystyle \left(0,-2\right)(0,−2).
The asymptote also shifts down 3 units to \displaystyle y=-3y=−3.
The range becomes \displaystyle \left(-3,\infty \right)(−3,∞).
Graphing a Horizontal Shift
The next transformation occurs when we add a constant c to the input of the parent function \displaystyle f\left(x\right)={b}^{x}f(x)=b
x
, giving us a horizontal shift c units in the opposite direction of the sign. For example, if we begin by graphing the parent function \displaystyle f\left(x\right)={2}^{x}f(x)=2
x
, we can then graph two horizontal shifts alongside it, using \displaystyle c=3c=3: the shift left, \displaystyle g\left(x\right)={2}^{x+3}g(x)=2
x+3
, and the shift right, \displaystyle h\left(x\right)={2}^{x - 3}h(x)=2
x−3
. Both horizontal shifts are shown in Figure 6.
Graph of three functions, g(x) = 2^(x+3) in blue, f(x) = 2^x in orange, and h(x)=2^(x-3). Each functions’ asymptotes are at y=0Note that each functions’ transformations are described in the text.
Figure 6
Observe the results of shifting \displaystyle f\left(x\right)={2}^{x}f(x)=2
x
horizontally:
The domain, \displaystyle \left(-\infty ,\infty \right)(−∞,∞), remains unchanged.
The asymptote, \displaystyle y=0y=0, remains unchanged.
The y-intercept shifts such that:
When the function is shifted left 3 units to \displaystyle g\left(x\right)={2}^{x+3}g(x)=2
x+3
, the y-intercept becomes \displaystyle \left(0,8\right)(0,8). This is because \displaystyle {2}^{x+3}=\left(8\right){2}^{x}2
x+3
=(8)2
x
, so the initial value of the function is 8.
When the function is shifted right 3 units to \displaystyle h\left(x\right)={2}^{x - 3}h(x)=2
x−3
, the y-intercept becomes \displaystyle \left(0,\frac{1}{8}\right)(0,
8
1
). Again, see that \displaystyle {2}^{x - 3}=\left(\frac{1}{8}\right){2}^{x}2
x−3
=(
8
1
)2
x
, so the initial value of the function is \displaystyle \frac{1}{8}
8
1
.
A GENERAL NOTE: SHIFTS OF THE PARENT FUNCTION \DISPLAYSTYLE F\LEFT(X\RIGHT)={B}^{X}F(X)=B
X
For any constants c and d, the function \displaystyle f\left(x\right)={b}^{x+c}+df(x)=b
x+c
+d shifts the parent function \displaystyle f\left(x\right)={b}^{x}f(x)=b
x
vertically d units, in the same direction of the sign of d.
horizontally c units, in the opposite direction of the sign of c.
The y-intercept becomes \displaystyle \left(0,{b}^{c}+d\right)(0,b
c
+d).
The horizontal asymptote becomes y = d.
The range becomes \displaystyle \left(d,\infty \right)(d,∞).
The domain, \displaystyle \left(-\infty ,\infty \right)(−∞,∞), remains unchanged.
HOW TO: GIVEN AN EXPONENTIAL FUNCTION WITH THE FORM \DISPLAYSTYLE F\LEFT(X\RIGHT)={B}^{X+C}+DF(X)=B
X+C
+D, GRAPH THE TRANSLATION.
Draw the horizontal asymptote y = d.
Identify the shift as \displaystyle \left(-c,d\right)(−c,d). Shift the graph of \displaystyle f\left(x\right)={b}^{x}f(x)=b
x
left c units if c is positive, and right \displaystyle cc units if c is negative.
Shift the graph of \displaystyle f\left(x\right)={b}^{x}f(x)=b
x
up d units if d is positive, and down d units if d is negative.
State the domain, \displaystyle \left(-\infty ,\infty \right)(−∞,∞), the range, \displaystyle \left(d,\infty \right)(d,∞), and the horizontal asymptote \displaystyle y=dy=d.
Step-by-step explanation:
