We want to write the exponential equation of the parent function represented by the graph. Then we will write a function for the indicated translation. Let's approach these tasks one at a time.

Writing the Equation for the Parent Function

Exponential functions all follow the same general form.

y = a b^{x}

By identifying two points on the given graph, we can use them and the general form shown above to determine the equation of parent function.

The points

(0, 4)

and

(1, 2)

both lie on the curve. Notice that

(0, 4)

is the

y -

intercept. We can substitute the

y -

intercept into the formula to solve for

a .

y = a b^{x}

SubstituteII

$x = 0$ , $y = 4$

4 = a b^{0}

▼

Solve for

a

$b^{0} = 1$

4 = a (1)

IdPropMult

Identity Property of Multiplication

4 = a

RearrangeEqn

Rearrange equation

a = 4

Having found

a,

we can substitute this into the general form.

y = 4 b^{x}

Next, we can substitute

x = 1

and

y = 2

into the updated equation to solve for

b .

y = 4 b^{x}

SubstituteII

$x = 1$ , $y = 2$

2 = 4 b^{1}

▼

Solve for

b

$b^{1} = b$

2 = 4 b

DivEqn

$LHS / 4 = RHS / 4$

\frac{1}{2} = b

RearrangeEqn

Rearrange equation

b = \frac{1}{2}

Finally, we can write the parent function represented by the graph.

y = 4 (\frac{1}{2})^{x}

Function for the Indicated Translation

Now, we want to write a function for the indicated translation.

left 4 units and up 3 units

To do so, we should recall two things.

If a horizontal translation is to the right, we subtract from the $x -$ variable. If the translation is to the left, we add to the $x -$ variable.
If a vertical translation is up, we add to the $y -$ variable. If the translation is down, we subtract from the $y -$ variable.

Therefore, for the indicated translation, we have to

add 4

to the

x -

variable and

add 3

to the entire function.

y = 4 (\frac{1}{2})^{x + 4} + 3

Exercise