Question

write an equation for the function graphed below

(y =)

Explanation:

⚡ Using what you learned: Graphing Rational Functions · Identifying and Analyzing Asymptotes

Step 1: Identify asymptotes and intercepts

Vertical asymptotes:

$$ x = -2, \quad x = 3 $$

Horizontal asymptote:

$$ y = 0 $$

\(x\)-intercept:

$$ (2, 0) $$

\(y\)-intercept:

$$ (0, 2) $$

Step 2: Formulate the rational function template

Using the vertical asymptotes and the \(x\)-intercept:

$$ y = \frac{a(x - 2)}{(x + 2)(x - 3)} $$

Step 3: Solve for the constant \(a\)

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

$$ 2 = \frac{a(0 - 2)}{(0 + 2)(0 - 3)} $$

$$ 2 = \frac{-2a}{-6} $$

$$ 2 = \frac{a}{3} $$

$$ a = 6 $$

Substitute \(a = 6\) back into the equation:

$$ y = \frac{6(x - 2)}{(x + 2)(x - 3)} $$

Answer:

$$ y = \frac{6(x - 2)}{(x + 2)(x - 3)} $$