Consider the function [tex]f(x)=x^2+12 x+11[/tex].
[tex]x[/tex]-intercepts:
[tex]
\begin{array}{l}
0 = x ^2+ 1 2 x+11 \\
0=(x+1)(x+11)
\end{array}
[/tex]
[tex]y[/tex]-intercept:
[tex]f(0)=(0)^2+12(0)+11[/tex]
What are the intercepts of the function?
The [tex]x[/tex]-intercepts are $\square$
The [tex]y[/tex]-intercept is $\square$
Answer (1)
Find the x-intercepts by setting f ( x ) = 0 and solving for x : x 2 + 12 x + 11 = ( x + 1 ) ( x + 11 ) = 0 , which gives x = − 1 and x = − 11 .
Find the y-intercept by setting x = 0 and evaluating f ( 0 ) : f ( 0 ) = ( 0 ) 2 + 12 ( 0 ) + 11 = 11 .
The x-intercepts are -1 and -11.
The y-intercept is 11, so the final answer is: x = − 1 , − 11 ; y = 11
Explanation
Understanding the Problem We are given the function f ( x ) = x 2 + 12 x + 11 and asked to find its x and y intercepts. The x -intercepts are the points where the graph of the function intersects the x -axis, which means f ( x ) = 0 . The y -intercept is the point where the graph intersects the y -axis, which means x = 0 .
Finding the x-intercepts To find the x -intercepts, we need to solve the equation x 2 + 12 x + 11 = 0 . The problem already provides the factored form of the quadratic: ( x + 1 ) ( x + 11 ) = 0 . This means that either x + 1 = 0 or x + 11 = 0 . Solving these equations gives us x = − 1 or x = − 11 . So the x -intercepts are − 1 and − 11 .
Finding the y-intercept To find the y -intercept, we need to evaluate f ( 0 ) . The problem provides the calculation: f ( 0 ) = ( 0 ) 2 + 12 ( 0 ) + 11 = 0 + 0 + 11 = 11 . So the y -intercept is 11 .
Final Answer Therefore, the x -intercepts are − 1 and − 11 , and the y -intercept is 11 .
Examples
Understanding intercepts is crucial in various real-world applications. For example, in business, the x-intercept can represent the break-even point where costs equal revenue. The y-intercept can represent the initial investment or fixed costs before any units are sold. By analyzing the intercepts of a function, businesses can make informed decisions about pricing, production, and investment strategies. Similarly, in physics, intercepts can represent initial conditions or equilibrium points in a system.
