Let's solve for the intercepts of the function \( f(x) = x^2 + 12x + 11 \).
1. Finding the \( x \)-intercepts:
To find the \( x \)-intercepts, we need to solve the equation \( f(x) = 0 \).
[tex]\[
0 = x^2 + 12x + 11
\][/tex]
Factor the quadratic equation.
[tex]\[
0 = (x + 1)(x + 11)
\][/tex]
Set each factor equal to zero to solve for the \( x \)-intercepts:
[tex]\[
x + 1 = 0 \quad \Rightarrow \quad x = -1
\][/tex]
[tex]\[
x + 11 = 0 \quad \Rightarrow \quad x = -11
\][/tex]
Therefore, the \( x \)-intercepts are \( x = -1 \) and \( x = -11 \).
2. Finding the \( y \)-intercept:
To find the \( y \)-intercept, we need to evaluate the function at \( x = 0 \).
[tex]\[
f(0) = (0)^2 + 12(0) + 11
\][/tex]
[tex]\[
f(0) = 11
\][/tex]
Therefore, the \( y \)-intercept is 11.
Thus, the intercepts of the function are:
- The \( x \)-intercepts are \( -1 \) and \( -11 \).
- The [tex]\( y \)[/tex]-intercept is 11.
