Question

practice & problem solving

1. make sense and persevere let ( f(x) = a^x ). describe two ways you could identify the value of ( k ) in the transformation implied by ( g(x) = a^x + k ) from the graphs of ( f ) and ( g ).
2. error analysis describe and correct the error a student made in analyzing the transformation ( g(x) = a^{x - h} ).

the graph of ( g(x) = a^{x - h} ) is the graph of ( f(x) = a^x ) translated ( h ) units to the left. (\boldsymbol{\times})

1. higher order thinking think about the graph of ( f(x) = 2^x ).

a. compare the graph of ( g(x) = 2^{x + 3} + 4 ) to the graph of ( f(x) = 2^x ).
b. in general, when the graph of an exponential function is translated both vertically and horizontally, what is the effect on the asymptote?
c. in general, when the graph of an exponential function is translated both vertically and horizontally, what is the effect on the domain and the range?

1. a. use appropriate tools copy and complete the table. compare the graphs of ( f ) and ( g ).

( x )	( f(x) = 4^{\frac{1}{2}x} )	( g(x) = 4^x )
( -1 )
( 0 )
( 2 )
( 4 )

b. what point do the functions have in common?
c. describe the asymptote of each function.
practice
compare the graph of each function to the graph of ( f(x) = 2^x ). see examples 1–3

1. ( g(x) = 2^x - 6 )
2. ( p(x) = 2^{x + 4} )
3. ( g(x) = 2^{x - 1} )
4. ( j(x) = 2^x + \frac{3}{4} )

find the value of ( k ) or ( h ) in each of the graphs. see examples 1–3
23.
graph: ( g(x) = 3^x + k ) and ( f(x) = 3^x ) on a coordinate plane with x from -4 to 4 and y from 0 to 6
24.
graph: ( f(x) = 3^x ) and ( g(x) = 3^{x - h} ) on a coordinate plane with x from -4 to 4 and y from 0 to 8
graph each function and its transformation. see examples 1–3

1. ( f(x) = 4^x )

( g(x) = 4^x + k ) for ( k = -4 )

1. ( f(x) = 0.5^x )

( g(x) = 0.5^{x - h} ) for ( h = -5 )
compare the y-intercepts, asymptotes, and ranges for the graphs of ( f ) and ( g ). see example 3

1. ( f(x) = 4^x )

( x )	( g(x) )
( -1 )	3.25
( 0 )	4
( 1 )	7
( 2 )	19

28.
graph: the graph of ( g ) is a horizontal translation 3 units to the left of the graph of ( f(x) = 2^x ).

Explanation:

Step1: Solve 19. $g(x)=2^x -6$

Compare to $f(x)=2^x$: vertical shift down 6 units.

Step2: Solve 20. $p(x)=2^{x+4}$

Rewrite as $2^{x-(-4)}$: horizontal shift left 4 units.

Step3: Solve 21. $g(x)=2^{x-1}$

Compare to $f(x)=2^x$: horizontal shift right 1 unit.

Step4: Solve 22. $j(x)=2^x + \frac{3}{4}$

Compare to $f(x)=2^x$: vertical shift up $\frac{3}{4}$ units.

Step5: Solve 23. Find $k$

$f(x)=3^x$, $g(x)=3^x +k$. At $x=0$, $f(0)=1$, $g(0)=3$. So $3=1+k$, $k=2$.

Step6: Solve 24. Find $h$

$f(x)=3^x$, $g(x)=3^{x-h}$. At $x=2$, $g(2)=3^{2-h}=3^0=1$, so $2-h=0$, $h=2$.

Step7: Solve 17a. $g(x)=2^{x+3}+4$

Rewrite as $2^{x-(-3)}+4$: shift left 3, up 4 from $f(x)=2^x$.

Step8: Solve 17b. Asymptote effect

Parent asymptote $y=0$. Vertical shift up 4: new asymptote $y=4$.

Step9: Solve 17c. Domain/Range effect

Domain remains all real numbers. Parent range $(0,\infty)$; new range $(4,\infty)$.

Step10: Solve 18a. Complete table

For $f(x)=4^{\frac{1}{2}x}=(4^{\frac{1}{2}})^x=2^x$:
$x=-2$: $2^{-2}=\frac{1}{4}$; $x=-1$: $2^{-1}=\frac{1}{2}$; $x=0$: $2^0=1$; $x=2$: $2^2=4$; $x=4$: $2^4=16$.
For $g(x)=4^x$:
$x=-2$: $4^{-2}=\frac{1}{16}$; $x=-1$: $4^{-1}=\frac{1}{4}$; $x=0$: $4^0=1$; $x=2$: $4^2=16$; $x=4$: $4^4=256$.

Step11: Solve 18b. Common point

From table, both have $(0,1)$.

Step12: Solve 18c. Asymptotes

Both $f(x)=2^x$ and $g(x)=4^x$ have horizontal asymptote $y=0$.

Step13: Solve 25. Graph $f(x)=4^x$, $g(x)=4^x-4$

$g(x)$ is $f(x)$ shifted down 4; asymptote $y=-4$ for $g$, $y=0$ for $f$.

Step14: Solve 26. Graph $f(x)=0.5^x$, $g(x)=0.5^{x-(-5)}=0.5^{x+5}$

$g(x)$ is $f(x)$ shifted left 5 units; same asymptote $y=0$.

Step15: Solve 27. $f(x)=4^x$, $g(x)$ table

$g(x)=4^x + 3$ (check $x=0$: $4^0+3=4$ matches). Y-intercept: $f(0)=1$, $g(0)=4$. Asymptotes: $y=0$ (f), $y=3$ (g). Ranges: $(0,\infty)$ (f), $(3,\infty)$ (g).

Step16: Solve 28. $g(x)=2^{x+3}$

Y-intercept: $g(0)=8$, $f(0)=1$. Asymptotes: $y=0$ for both. Ranges: $(0,\infty)$ for both.

Step17: Solve 16. Error correction

Student said $g(x)=a^{x-h}$ shifts left $h$; correct: shifts right $h$ units.

Answer:

1. The graph of $g(x)$ is the graph of $f(x)=2^x$ translated 6 units down.
2. The graph of $p(x)$ is the graph of $f(x)=2^x$ translated 4 units left.
3. The graph of $g(x)$ is the graph of $f(x)=2^x$ translated 1 unit right.
4. The graph of $j(x)$ is the graph of $f(x)=2^x$ translated $\frac{3}{4}$ units up.
5. $k=2$
6. $h=2$

17a. The graph of $g(x)$ is the graph of $f(x)=2^x$ translated 3 units left and 4 units up.
17b. The horizontal asymptote is translated vertically by the same amount as the function; for $g(x)=2^{x+3}+4$, the asymptote shifts from $y=0$ to $y=4$.
17c. The domain remains all real numbers. The range of the parent function $(0,\infty)$ shifts vertically to $(4,\infty)$ for $g(x)$.
18a.

$x$	$f(x)=4^{\frac{1}{2}x}$	$g(x)=4^x$
$-1$	$\frac{1}{2}$	$\frac{1}{4}$
$0$	$1$	$1$
$2$	$4$	$16$
$4$	$16$	$256$

18b. The functions share the point $(0, 1)$.
18c. Both functions have the horizontal asymptote $y=0$.

1. $f(x)=4^x$ is an increasing exponential graph with asymptote $y=0$; $g(x)=4^x-4$ is the same graph shifted 4 units down, with asymptote $y=-4$.
2. $f(x)=0.5^x$ is a decreasing exponential graph with asymptote $y=0$; $g(x)=0.5^{x+5}$ is the same graph shifted 5 units left, with asymptote $y=0$.
3. Y-intercepts: $f(x)$ has $(0,1)$, $g(x)$ has $(0,4)$. Asymptotes: $y=0$ for $f(x)$, $y=3$ for $g(x)$. Ranges: $(0, \infty)$ for $f(x)$, $(3, \infty)$ for $g(x)$.
4. $g(x)=2^{x+3}$. Y-intercepts: $f(x)$ has $(0,1)$, $g(x)$ has $(0,8)$. Asymptotes: $y=0$ for both functions. Ranges: $(0, \infty)$ for both functions.
5. The student's error is stating $g(x)=a^{x-h}$ is a left shift; the correct transformation is a horizontal shift $h$ units to the right.