Consider the function [tex]f(x)=x^2+12 x+11[/tex]
x-intercepts:
[tex]
\begin{array}{l}
0=x^2+12 x+11 \\
0=(x+1)(x+11)
\end{array}
[/tex]
y-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)
To find the x-intercepts, set f ( x ) = 0 and solve for x . Factoring the quadratic gives ( x + 1 ) ( x + 11 ) = 0 , so the x-intercepts are x = − 1 and x = − 11 .
To find the y-intercept, evaluate f ( 0 ) . Substituting x = 0 into the function gives 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 quadratic 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 occurs when 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 . Setting each factor equal to zero gives us the solutions x + 1 = 0 or x + 11 = 0 . Solving for x , we get x = − 1 or x = − 11 . Thus, the x -intercepts are − 1 and − 11 .
Finding the y-intercept To find the y -intercept, we need to evaluate f ( 0 ) . The problem states f ( 0 ) = ( 0 ) 2 + 12 ( 0 ) + 11 . This simplifies to f ( 0 ) = 0 + 0 + 11 = 11 . Thus, the y -intercept is 11 .
Final Answer The x -intercepts are − 1 and − 11 , and the y -intercept is 11 .
Examples
Understanding intercepts is crucial in various real-world applications. For instance, in business, the x-intercept can represent the break-even point where costs equal revenue. If we model profit as a function of sales, the x-intercept tells us the sales volume needed to start making a profit. Similarly, the y-intercept can represent initial costs or investments before any sales are made. Analyzing these intercepts helps businesses make informed decisions about pricing, production, and investment strategies.
