This graph shows an exponential function.

Question

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:

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