Question

This graph shows an exponential function.
Write an equation for the function in the form f(x)=a(b)x.
f(x)=
graph points: (1,-12)
(2,-48)

Answer 1

Given that the function is exponential and in the form \( f(x) = a(b)^x \), we can use the points you provided, \((1, -12)\) and \((2, -48)\), to find the values of \( a \) and \( b \).

1. Substitute the first point \((1, -12)\) into the equation:
\[
f(1) = a(b)^1 = -12
\]
This simplifies to:
\[
ab = -12 \quad \text{(Equation 1)}
\]

2. Substitute the second point \((2, -48)\) into the equation:
\[
f(2) = a(b)^2 = -48
\]
This simplifies to:
\[
ab^2 = -48 \quad \text{(Equation 2)}
\]

Now we have two equations:
1. \( ab = -12 \)
2. \( ab^2 = -48 \)

Now, we can divide Equation 2 by Equation 1 to eliminate \( a \):
\[
\frac{ab^2}{ab} = \frac{-48}{-12}
\]
This simplifies to:
\[
b = 4
\]

Now that we have \( b = 4 \), we can substitute this value back into Equation 1 to find \( a \):
\[
a(4) = -12
\]
Thus:
\[
a = -3
\]

Now we have both values:
- \( a = -3 \)
- \( b = 4 \)

Finally, we can write the equation of the function:
\[
f(x) = -3(4)^x
\]

Thus, the equation for the exponential function is:
\[
f(x) = -3(4)^x
\]

Answer 2

Write an equation for the sequence below. Use n to represent the position of a term in the sequence, where n=1 for the first term.
–
3,
–
15,
–
75,
–
375, ...
an=

Answer 3

To find an equation for the given sequence \( -3, -15, -75, -375, \ldots \), we first observe the pattern of the terms.

The sequence starts with:
- \( a_1 = -3 \)
- \( a_2 = -15 \)
- \( a_3 = -75 \)
- \( a_4 = -375 \)

Next, we note the relationship between the terms. It looks like each term is being multiplied by 5 to get the next term:

\[
\begin{align*}
a_1 &= -3 \\
a_2 &= -3 \times 5 = -15 \\
a_3 &= -15 \times 5 = -75 \\
a_4 &= -75 \times 5 = -375
\end{align*}
\]

This indicates that the sequence is geometric with a common ratio \( r = 5 \).

### General Formula for a Geometric Sequence

The general formula for a geometric sequence is given by:

\[
a_n = a_1 \cdot r^{(n-1)}
\]

Where:
- \( a_n \) is the \( n \)-th term,
- \( a_1 \) is the first term,
- \( r \) is the common ratio,
- \( n \) is the position of the term in the sequence.

### Applying the values

Here, \( a_1 = -3 \) and \( r = 5 \). Substituting these values into the formula gives:

\[
a_n = -3 \cdot 5^{(n-1)}
\]

### Final Result

Thus, the equation for the sequence is:

\[
\boxed{a_n = -3 \cdot 5^{(n-1)}}
\]