Shifting Functions Math Example 2

Follow the full solution, then compare it with the other examples linked below.

Example 2

medium

The graph of

f(x)=e^x

is shifted left

2

units and down

5

units to give

g(x)

. Write

g(x)

, find

g(0)

, and determine if the horizontal asymptote changes.

Solution

1
Left $2$ : replace $x$ with $x+2$ → $e^{x+2}$ . Down $5$ : subtract $5$ → $g(x)=e^{x+2}-5$ .
2
Evaluate: $g(0)=e^{0+2}-5=e^2-5\approx7.389-5=2.389$ .
3
Horizontal asymptote of $f$ : $y=0$ . After shifting down $5$ : the asymptote shifts to $y=-5$ . So $g(x)\to-5$ as $x\to-\infty$ .

Answer

g(x)=e^{x+2}-5

;

g(0)=e^2-5\approx2.39

; asymptote shifts to

y=-5

Vertical shifts move the entire graph, including asymptotes. Shifting down

5

moves the horizontal asymptote of

e^x

from

y=0

to

y=-5

. The function still grows without bound as

x\to+\infty

.

About Shifting Functions

Shifting a function translates its graph horizontally or vertically without changing its shape: $f(x - h) + k$ shifts right by $h$ and up by $k$ .

Learn more about Shifting Functions →

More Shifting Functions Examples

Starting from [formula], describe and sketch the transformations for [formula].

The point [formula] is on the graph of [formula]. Find the corresponding point on each shifted graph

Example 4 medium

Write the equation of the function whose graph is [formula] shifted right [formula] and reflected ov

← All Shifting Functions Examples Shifting Functions Concept

Related Concepts

Function Transformation