Question

the graph below shows a transformation of (y = 2^x)

write an equation for the graph above.

(y =)

Explanation:

⚡ Using what you learned: Transformations of Exponential Functions

Step 1: Identify the horizontal asymptote

The graph flattens out as \( x \to -\infty \). Looking at the grid, the curve approaches the horizontal line \( y = 3 \).

This means the vertical shift is \( +3 \). The general equation is:

$$ y = a \cdot 2^{b \cdot x} + 3 $$

or, assuming a standard reflection and shift:

$$ y = a \cdot 2^{x} + 3 \quad \text{or} \quad y = -2^{x-h} + k $$

Let's test the form:

$$ y = -2^{x-h} + 3 $$

Step 2: Use key points to find the horizontal shift

We can identify clear integer coordinate points on the graph:

• \( (0, -1) \)
• \( (-1, 1) \)
• \( (-2, 2) \)
• \( (1, -5) \)

Let's substitute the point \( (0, -1) \) into \( y = -2^{x-h} + 3 \):

$$ -1 = -2^{0-h} + 3 $$

$$ -4 = -2^{-h} $$

$$ 4 = 2^{-h} $$

$$ 2^2 = 2^{-h} $$

$$ -h = 2 \implies h = -2 $$

Step 3: Verify the equation with another point

Substituting \( h = -2 \) back into our model gives:

$$ y = -2^{x - (-2)} + 3 $$

$$ y = -2^{x+2} + 3 $$

Let's verify with the point \( (-1, 1) \):

$$ y = -2^{-1+2} + 3 = -2^{1} + 3 = -2 + 3 = 1 $$

This matches the graph perfectly.

Let's verify with the point \( (1, -5) \):

$$ y = -2^{1+2} + 3 = -2^{3} + 3 = -8 + 3 = -5 $$

This also matches the graph perfectly.

Answer:

$$ y = -2^{x+2} + 3 $$