# 6.2 Graphs of exponential functions  (Page 3/6)

 Page 3 / 6

## Shifts of the parent function f ( x ) = b x

For any constants $\text{\hspace{0.17em}}c\text{\hspace{0.17em}}$ and $\text{\hspace{0.17em}}d,$ the function $\text{\hspace{0.17em}}f\left(x\right)={b}^{x+c}+d\text{\hspace{0.17em}}$ shifts the parent function $\text{\hspace{0.17em}}f\left(x\right)={b}^{x}$

• vertically $\text{\hspace{0.17em}}d\text{\hspace{0.17em}}$ units, in the same direction of the sign of $\text{\hspace{0.17em}}d.$
• horizontally $\text{\hspace{0.17em}}c\text{\hspace{0.17em}}$ units, in the opposite direction of the sign of $\text{\hspace{0.17em}}c.$
• The y -intercept becomes $\text{\hspace{0.17em}}\left(0,{b}^{c}+d\right).$
• The horizontal asymptote becomes $\text{\hspace{0.17em}}y=d.$
• The range becomes $\text{\hspace{0.17em}}\left(d,\infty \right).$
• The domain, $\text{\hspace{0.17em}}\left(-\infty ,\infty \right),$ remains unchanged.

Given an exponential function with the form $\text{\hspace{0.17em}}f\left(x\right)={b}^{x+c}+d,$ graph the translation.

1. Draw the horizontal asymptote $\text{\hspace{0.17em}}y=d.$
2. Identify the shift as $\text{\hspace{0.17em}}\left(-c,d\right).\text{\hspace{0.17em}}$ Shift the graph of $\text{\hspace{0.17em}}f\left(x\right)={b}^{x}\text{\hspace{0.17em}}$ left $\text{\hspace{0.17em}}c\text{\hspace{0.17em}}$ units if $\text{\hspace{0.17em}}c\text{\hspace{0.17em}}$ is positive, and right $\text{\hspace{0.17em}}c\text{\hspace{0.17em}}$ units if $c\text{\hspace{0.17em}}$ is negative.
3. Shift the graph of $\text{\hspace{0.17em}}f\left(x\right)={b}^{x}\text{\hspace{0.17em}}$ up $\text{\hspace{0.17em}}d\text{\hspace{0.17em}}$ units if $\text{\hspace{0.17em}}d\text{\hspace{0.17em}}$ is positive, and down $\text{\hspace{0.17em}}d\text{\hspace{0.17em}}$ units if $\text{\hspace{0.17em}}d\text{\hspace{0.17em}}$ is negative.
4. State the domain, $\text{\hspace{0.17em}}\left(-\infty ,\infty \right),$ the range, $\text{\hspace{0.17em}}\left(d,\infty \right),$ and the horizontal asymptote $\text{\hspace{0.17em}}y=d.$

## Graphing a shift of an exponential function

Graph $\text{\hspace{0.17em}}f\left(x\right)={2}^{x+1}-3.\text{\hspace{0.17em}}$ State the domain, range, and asymptote.

We have an exponential equation of the form $\text{\hspace{0.17em}}f\left(x\right)={b}^{x+c}+d,$ with $\text{\hspace{0.17em}}b=2,$ $\text{\hspace{0.17em}}c=1,$ and $\text{\hspace{0.17em}}d=-3.$

Draw the horizontal asymptote $\text{\hspace{0.17em}}y=d$ , so draw $\text{\hspace{0.17em}}y=-3.$

Identify the shift as $\text{\hspace{0.17em}}\left(-c,d\right),$ so the shift is $\text{\hspace{0.17em}}\left(-1,-3\right).$

Shift the graph of $\text{\hspace{0.17em}}f\left(x\right)={b}^{x}\text{\hspace{0.17em}}$ left 1 units and down 3 units.

The domain is $\text{\hspace{0.17em}}\left(-\infty ,\infty \right);\text{\hspace{0.17em}}$ the range is $\text{\hspace{0.17em}}\left(-3,\infty \right);\text{\hspace{0.17em}}$ the horizontal asymptote is $\text{\hspace{0.17em}}y=-3.$

Graph $\text{\hspace{0.17em}}f\left(x\right)={2}^{x-1}+3.\text{\hspace{0.17em}}$ State domain, range, and asymptote.

The domain is $\text{\hspace{0.17em}}\left(-\infty ,\infty \right);\text{\hspace{0.17em}}$ the range is $\text{\hspace{0.17em}}\left(3,\infty \right);\text{\hspace{0.17em}}$ the horizontal asymptote is $\text{\hspace{0.17em}}y=3.$

Given an equation of the form $\text{\hspace{0.17em}}f\left(x\right)={b}^{x+c}+d\text{\hspace{0.17em}}$ for $\text{\hspace{0.17em}}x,$ use a graphing calculator to approximate the solution.

• Press [Y=] . Enter the given exponential equation in the line headed “ Y 1 = ”.
• Enter the given value for $\text{\hspace{0.17em}}f\left(x\right)\text{\hspace{0.17em}}$ in the line headed “ Y 2 = ”.
• Press [WINDOW] . Adjust the y -axis so that it includes the value entered for “ Y 2 = ”.
• Press [GRAPH] to observe the graph of the exponential function along with the line for the specified value of $\text{\hspace{0.17em}}f\left(x\right).$
• To find the value of $\text{\hspace{0.17em}}x,$ we compute the point of intersection. Press [2ND] then [CALC] . Select “intersect” and press [ENTER] three times. The point of intersection gives the value of x for the indicated value of the function.

## Approximating the solution of an exponential equation

Solve $\text{\hspace{0.17em}}42=1.2{\left(5\right)}^{x}+2.8\text{\hspace{0.17em}}$ graphically. Round to the nearest thousandth.

Press [Y=] and enter $\text{\hspace{0.17em}}1.2{\left(5\right)}^{x}+2.8\text{\hspace{0.17em}}$ next to Y 1 =. Then enter 42 next to Y2= . For a window, use the values –3 to 3 for $\text{\hspace{0.17em}}x\text{\hspace{0.17em}}$ and –5 to 55 for $\text{\hspace{0.17em}}y.\text{\hspace{0.17em}}$ Press [GRAPH] . The graphs should intersect somewhere near $\text{\hspace{0.17em}}x=2.$

For a better approximation, press [2ND] then [CALC] . Select [5: intersect] and press [ENTER] three times. The x -coordinate of the point of intersection is displayed as 2.1661943. (Your answer may be different if you use a different window or use a different value for Guess? ) To the nearest thousandth, $\text{\hspace{0.17em}}x\approx 2.166.$

Solve $\text{\hspace{0.17em}}4=7.85{\left(1.15\right)}^{x}-2.27\text{\hspace{0.17em}}$ graphically. Round to the nearest thousandth.

$x\approx -1.608$

## Graphing a stretch or compression

While horizontal and vertical shifts involve adding constants to the input or to the function itself, a stretch or compression occurs when we multiply the parent function $\text{\hspace{0.17em}}f\left(x\right)={b}^{x}\text{\hspace{0.17em}}$ by a constant $\text{\hspace{0.17em}}|a|>0.\text{\hspace{0.17em}}$ For example, if we begin by graphing the parent function $\text{\hspace{0.17em}}f\left(x\right)={2}^{x},$ we can then graph the stretch, using $\text{\hspace{0.17em}}a=3,$ to get $\text{\hspace{0.17em}}g\left(x\right)=3{\left(2\right)}^{x}\text{\hspace{0.17em}}$ as shown on the left in [link] , and the compression, using $\text{\hspace{0.17em}}a=\frac{1}{3},$ to get $\text{\hspace{0.17em}}h\left(x\right)=\frac{1}{3}{\left(2\right)}^{x}\text{\hspace{0.17em}}$ as shown on the right in [link] .

bsc F. y algebra and trigonometry pepper 2
given that x= 3/5 find sin 3x
4
DB
remove any signs and collect terms of -2(8a-3b-c)
-16a+6b+2c
Will
Joeval
(x2-2x+8)-4(x2-3x+5)
sorry
Miranda
x²-2x+9-4x²+12x-20 -3x²+10x+11
Miranda
x²-2x+9-4x²+12x-20 -3x²+10x+11
Miranda
(X2-2X+8)-4(X2-3X+5)=0 ?
master
The anwser is imaginary number if you want to know The anwser of the expression you must arrange The expression and use quadratic formula To find the answer
master
The anwser is imaginary number if you want to know The anwser of the expression you must arrange The expression and use quadratic formula To find the answer
master
Y
master
master
Soo sorry (5±Root11* i)/3
master
Mukhtar
explain and give four example of hyperbolic function
What is the correct rational algebraic expression of the given "a fraction whose denominator is 10 more than the numerator y?
y/y+10
Mr
Find nth derivative of eax sin (bx + c).
Find area common to the parabola y2 = 4ax and x2 = 4ay.
Anurag
A rectangular garden is 25ft wide. if its area is 1125ft, what is the length of the garden
to find the length I divide the area by the wide wich means 1125ft/25ft=45
Miranda
thanks
Jhovie
What do you call a relation where each element in the domain is related to only one value in the range by some rules?
A banana.
Yaona
given 4cot thither +3=0and 0°<thither <180° use a sketch to determine the value of the following a)cos thither
what are you up to?
nothing up todat yet
Miranda
hi
jai
hello
jai
Miranda Drice
jai
aap konsi country se ho
jai
which language is that
Miranda
I am living in india
jai
good
Miranda
what is the formula for calculating algebraic
I think the formula for calculating algebraic is the statement of the equality of two expression stimulate by a set of addition, multiplication, soustraction, division, raising to a power and extraction of Root. U believe by having those in the equation you will be in measure to calculate it
Miranda
state and prove Cayley hamilton therom
hello
Propessor
hi
Miranda
the Cayley hamilton Theorem state if A is a square matrix and if f(x) is its characterics polynomial then f(x)=0 in another ways evey square matrix is a root of its chatacteristics polynomial.
Miranda
hi
jai
hi Miranda
jai
thanks
Propessor
welcome
jai
What is algebra
algebra is a branch of the mathematics to calculate expressions follow.
Miranda
Miranda Drice would you mind teaching me mathematics? I think you are really good at math. I'm not good at it. In fact I hate it. 😅😅😅
Jeffrey
lolll who told you I'm good at it
Miranda
something seems to wispher me to my ear that u are good at it. lol
Jeffrey
lolllll if you say so
Miranda
but seriously, Im really bad at math. And I hate it. But you see, I downloaded this app two months ago hoping to master it.
Jeffrey
which grade are you in though
Miranda
oh woww I understand
Miranda
Jeffrey
Jeffrey
Miranda
how come you finished in college and you don't like math though
Miranda
gotta practice, holmie
Steve
if you never use it you won't be able to appreciate it
Steve
I don't know why. But Im trying to like it.
Jeffrey
yes steve. you're right
Jeffrey
so you better
Miranda
what is the solution of the given equation?
which equation
Miranda
I dont know. lol
Jeffrey
Miranda
Jeffrey