Question

the graph represents an exponential function, \\(v(x)\\).

write an equation to represent \\(v(x)\\).

Explanation:

Identify key features from the graph

We need to find the equation of the exponential function \(v(x)\).
Looking at the graph:

• The horizontal asymptote is the x-axis, which is \(y = 0\).
• The y-intercept is at \((0, -2)\).
• Another clear integer point on the curve is \((1, -4)\).

Determine the general form of the function

Since the horizontal asymptote is \(y = 0\), the exponential function has the general form:

$$v(x) = a \cdot b^x$$

Solve for the initial value \(a\)

Using the y-intercept \((0, -2)\):

$$v(0) = a \cdot b^0 = -2$$

$$a = -2$$

Solve for the base \(b\)

Using the point \((1, -4)\) and \(a = -2\):

$$v(1) = -2 \cdot b^1 = -4$$

$$b = 2$$

Write the final equation

Substituting \(a = -2\) and \(b = 2\) back into the general form:

$$v(x) = -2 \cdot 2^x$$

Answer:

\(v(x) = -2(2)^x\)