Question

write an equation for the function graphed below

\\(y =\\)

Explanation:

🆕 New Concept Discovered: Rational Functions from Graphs
Finding equations using asymptotes and intercepts

Step 1: Identify asymptotes

The graph shows vertical asymptotes (dashed red lines) where the function is undefined:

$$ x = -3 \quad \text{and} \quad x = 2 $$

These correspond to factors in the denominator:

$$ (x + 3)(x - 2) $$

The horizontal asymptote is the \(x\)-axis:

$$ y = 0 $$

This indicates that the degree of the numerator is less than the degree of the denominator.

Step 2: Identify intercepts

The graph crosses the \(x\)-axis at the \(x\)-intercept:

$$ (1, 0) $$

This corresponds to a factor in the numerator:

$$ (x - 1) $$

The graph crosses the \(y\)-axis at the \(y\)-intercept:

$$ (0, -1) $$

Step 3: Set up the general equation

Using the factors identified, we write the general form of the rational function:

$$ y = \frac{a(x - 1)}{(x + 3)(x - 2)} $$

where \(a\) is a constant stretch factor.

Step 4: Solve for the constant factor

Substitute the \(y\)-intercept \((0, -1)\) into the equation to find \(a\):

$$ -1 = \frac{a(0 - 1)}{(0 + 3)(0 - 2)} $$

$$ -1 = \frac{-a}{(3)(-2)} $$

$$ -1 = \frac{-a}{-6} $$

$$ -1 = \frac{a}{6} $$

$$ a = -6 $$

Step 5: Write the final equation

Substitute \(a = -6\) back into the general form:

$$ y = \frac{-6(x - 1)}{(x + 3)(x - 2)} $$

Answer:

$$ y = \frac{-6(x - 1)}{(x + 3)(x - 2)} $$