Question

find an equation for the graph sketched below.

(f(x) =)

Explanation:

⚡ Using what you learned: Transformations of Exponential Functions

Step 1: Identify the general form and asymptote

The graph represents an exponential function that has been reflected and shifted.

Looking at the left side of the graph, the curve flattens out and approaches a horizontal asymptote at \( y = 6 \).

The general form of an exponential function with a horizontal asymptote at \( y = k \) is:

$$ f(x) = a \cdot b^x + k $$

Since the horizontal asymptote is \( y = 6 \):

$$ k = 6 $$

$$ f(x) = a \cdot b^x + 6 $$

Step 2: Use the y-intercept to find \( a \)

The graph crosses the y-axis at the point \( (0, 4) \). Substitute \( x = 0 \) and \( f(0) = 4 \) into the equation:

$$ 4 = a \cdot b^0 + 6 $$

$$ 4 = a(1) + 6 $$

$$ a = -2 $$

Now the equation is:

$$ f(x) = -2 \cdot b^x + 6 $$

Step 3: Use another clear point to find \( b \)

The graph passes clearly through the point \( (1, 2) \). Substitute \( x = 1 \) and \( f(1) = 2 \) into the equation:

$$ 2 = -2 \cdot b^1 + 6 $$

$$ 2 = -2b + 6 $$

$$ -4 = -2b $$

$$ b = 2 $$

Thus, the equation is:

$$ f(x) = -2 \cdot 2^x + 6 $$

Answer:

$$ f(x) = -2(2)^x + 6 $$