What is the $y$-intercept of the quadratic function $f(x)=(x-8)(x+3)$?
A. $(8,0)$
B. $(0,3)$
C. $(0,-24)$
D. $(-5,0)$
Answer (2)
To find the y -intercept, set x = 0 in the function.
Calculate f ( 0 ) = ( 0 − 8 ) ( 0 + 3 ) .
Simplify to find f ( 0 ) = − 24 .
The y -intercept is ( 0 , − 24 ) , so the answer is ( 0 , − 24 ) .
Explanation
Understanding the Problem We are given the quadratic function f ( x ) = ( x − 8 ) ( x + 3 ) and asked to find its y -intercept. The y -intercept is the point where the graph of the function intersects the y -axis. This occurs when x = 0 .
Calculating f(0) To find the y -intercept, we need to evaluate f ( 0 ) . We substitute x = 0 into the function: f ( 0 ) = ( 0 − 8 ) ( 0 + 3 ) = ( − 8 ) ( 3 ) = − 24
Determining the y-intercept The y -intercept is the point ( 0 , f ( 0 )) , which in this case is ( 0 , − 24 ) .
Examples
Understanding y-intercepts is crucial in many real-world applications. For example, in business, if you graph a cost function, the y-intercept represents the fixed costs, which are the costs incurred even when no units are produced. Similarly, in physics, if you graph the height of an object over time, the y-intercept represents the initial height of the object. Knowing how to find the y-intercept helps in interpreting the initial conditions or fixed values in various scenarios.
The y -intercept of the function f ( x ) = ( x − 8 ) ( x + 3 ) is found by evaluating f ( 0 ) , which gives f ( 0 ) = − 24 . Therefore, the y -intercept is the point ( 0 , − 24 ) , corresponding to option C. ( 0 , − 24 ) .
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