8.3 Exponential Functions

A function of the form y = bx , where b > 0, b ≠ 1, is called an exponential function of x.
The graph y = 2x, y = 3x, y = 2-x, and 3-x are as follows.
fig 8.3-1

fig 8.3-2

Domain of y = bx : ℝ
Range: x > 0
Asymptote: x-axis (y = 0)

Note that y = logbx and y = bx are inverse of each other.
For b > 1, y = bx is an exponential growth function as the functions
y = 2-x = (1 ⁄2)x and y = 3-x = (1 ⁄3)x

$$\small{y = b^x \xrightarrow[\text{on} \, y\text{-axis}] {\text{reflection}} \, y = b^{-x} }$$
From the graph of y = bx, the graph of y = abx , a > 0 can be obtained as
$$\small{y = b^x \xrightarrow[\text{scale factor} \,a] {\text{vertical scaling}} \, y = ab^{x} }$$
From the graph of y = abx, a > 0, the graph of y = abx - h + k can be obtained as
$$\small{y = ab^x \xrightarrow[\text{vertical translation} \, k\,\text{units}] {\text{horizontal translation}\, h\, \text{units}} \, y = ab^{x - h} + k }$$ Example 6.
Draw the graph of y = 1.5 ⋅ 2x-1 - 1 from y = 2x , and y = -2-(x+1) + 2 from y = -2-x.
Solution
$$\small{y = 2^x \xrightarrow[\text{scale factor} \,1.5] {\text{vertical scaling}} \, y }$$ $$\small{ = 1.5 \sdot 2^x \xrightarrow[\text{vertical translation -1 unit}] {\text{horizontal translation 1 unit}} \, y }$$ = 1.5 ⋅ 2x-1 - 1
fig 8.3-3

Note: Asymptote y = -1


$$\small{y = -2^{-x} \xrightarrow[\text{2 units}] {\text{vertical translation}} \, y }$$ $$\small{ = -2^{-x} + 2 \xrightarrow[\text{-1 unit}] {\text{horizontal translation}} \, y }$$ = -2-(x+1) + 2

fig 8.3-4

Note: Asymptote y = 2

Example 7.
Points (0,1) and (1, b) are on the graph of y = bx . Find the corresponding points on the graphs of y = abx and y = abx - h + k. What is the asymptote of y = abx-h + k ?
Find the range of y = abx-h + k if a > 0 and if a < 0.
Solutiion
$$\small{y = b^x \xrightarrow[\text{scale factor} \,a] {\text{vertical scaling}} \, y }$$ $$\small{ = ab^x \xrightarrow[\text{vertical translation}\, k \, \text{units}] {\text{horizontal translation}\, h\, \text{units}} \, y }$$ = abx-h + k
$$\small{(0, 1) \xrightarrow[\text{scale factor} \,a] {\text{vertical scaling}}\, (0,a) \xrightarrow[\text{vertical translation}\, k \, \text{units}] {\text{horizontal translation}}\, (h\,, a+k) }$$ $$\small{(1,b) \xrightarrow[\text{scale factor} \,a] {\text{vertical scaling}}\, (1,ab) \xrightarrow[\text{vertical translation}\, k \, \text{units}] {\text{horizontal translation}}\, (1+h\,, ab+k) }$$ Asymptote of y =bx is y = 0 (x-axis) and asymptote of y = abx is also y = 0 (x-axis).
After vertical translation k units, asymptote of y = abx-h + k is y = k.
If k > 0, than the range of y = abx-h + k is {y : y > k} that means all of y above the asymptote y = k.
If a < 0, then the range of y = abx-h + k is {y : y < k} that means all of y below the asymptote y = k.

Exercise 8.3
1. Draw the graph of
(a) y = 2 ⋅ 3x + 1 - 2
Solution
$$\small{y = 3^x \xrightarrow[\text{scale factor} \,2] {\text{vertical scaling}} \, y }$$ $$\small{ = 2 \sdot 3^x \xrightarrow[\text{vertical translation -2 units}] {\text{horizontal translation -1 unit}} \, y }$$ = 2 ⋅ 3x+1 -2
fig 8.3-5

Domain = R
Range {y:y > -2}

(b) y = -2x-1 + 3
Solution
$$\small{ = -2^x \xrightarrow[\text{vertical translation 3 units}] {\text{horizontal translation 1 unit}} \, y }$$ = -2x-1 + 3
fig 8.3-6

Domain = R
Range {y : y < 3}

2. Draw the graph of
(a) y = 2|x|
Solution
fig 8.3-7


Domain = R
Range {y : y > 1}

(b) y = 2|-x|
Solution
fig 8.3-8

Domain = R
Range {y : 0 < y < 1}

3. Find the y-intercept, asymptote and the range of
(a) y = 3ex-1 + 2
Solution
$$\small{y = e^x \xrightarrow[\text{scale factor} \,3] {\text{vertical scaling}} \, y }$$ $$\small{ = 3e^x \xrightarrow[\text{vertical translation}\, 2 \, \text{units}] {\text{horizontal translation}\, 1\, \text{unit}} \, 3e^{x-1} + 2 }$$ x = 0,
y = 3ex-1 + 2 = 3.10
asymptote y = 2
Domain = R
Range {y | y > 2}
fig 8.3-9


(b) y = -2e-x+1 + 3
Solution
$$\small{y = e^{-x} \xrightarrow[\text{scale factor} \,-2] {\text{vertical scaling}} \, y }$$ $$\small{ = -2e^{-x} \xrightarrow[\text{vertical translation}\, 3 \, \text{units}] {\text{horizontal translation}\, -1\, \text{unit}} \, -2e^{-x+1} + 3 }$$ x = 0,
y = -2e + 3 = -2.4
asymptote y = 3
Domain = R
Range {y | y < 3}
fig 8.3-10