Question

match the graph to the function

Explanation:

⚡ Using what you learned: Graphing Trigonometric Functions · understanding exponential functions

Step 1: Identify the first graph

The first graph shows a curve that starts very close to the horizontal axis on the left and grows rapidly as it moves to the right. It passes through the point \( (0, 2) \).

This is the characteristic shape of an exponential growth function of the form:

$$ y = a \cdot b^x $$

Specifically, when \( x = 0 \), \( y = 2 \), which corresponds to:

$$ y = 2 \cdot e^x \quad \text{or} \quad y = 2 \cdot 3^x $$

Step 2: Identify the second graph

The second graph has two separate branches with a vertical asymptote at \( x = 2 \) and a horizontal asymptote at \( y = 1 \).

This represents a rational function of the form:

$$ y = \frac{a}{x - h} + k $$

With asymptotes at \( x = 2 \) and \( y = 1 \), the function is:

$$ y = \frac{1}{x - 2} + 1 $$

Step 3: Identify the third graph

The third graph shows a repeating wave pattern that oscillates symmetrically above and below the horizontal axis.

This is a periodic trigonometric wave, specifically a sine or cosine function:

$$ y = A \sin(Bx) \quad \text{or} \quad y = A \cos(Bx) $$

Answer:

• First Graph (Top): Exponential Function (e.g., \( y = 2 \cdot 3^x \) or \( y = 2e^x \))
• Second Graph (Middle): Rational Function (e.g., \( y = \frac{1}{x - 2} + 1 \))
• Third Graph (Bottom): Trigonometric Function (e.g., \( y = 2\sin(x) \) or \( y = 2\cos(x) \))