To find the vertical asymptotes of the function \( f(x) = \frac{10}{x^2 – 7x – 30} \), we need to identify where the denominator equals zero, as vertical asymptotes occur at these values.
1. **Set the Denominator to Zero:** We begin by solving the equation:
\( x^2 – 7x – 30 = 0 \)
2. **Factor the Quadratic:** To solve for \( x \), we can factor the quadratic expression. We need two numbers that multiply to \( -30 \) (the constant term) and add up to \( -7 \) (the coefficient of the \( x \) term). The numbers \( -10 \) and \( 3 \) work because:
\( -10 \times 3 = -30 \) and \( -10 + 3 = -7 \).
Thus, we can factor the quadratic as:
\( (x – 10)(x + 3) = 0 \)
3. **Find the Roots:** Setting each factor equal to zero gives us:
\( x – 10 = 0 \Rightarrow x = 10 \
\( x + 3 = 0 \Rightarrow x = -3 \
4. **Conclusion:** The vertical asymptotes for the function \( f(x) \) occur at the points where the function is undefined, which are the roots we found. Therefore, the vertical asymptotes are:
- \( x = 10 \)
- \( x = -3 \)
In summary, the vertical asymptotes of the function \( f(x) = \frac{10}{x^2 – 7x – 30} \) are located at \( x = 10 \) and \( x = -3 \).