What is the $y$-intercept of the graph of the function $f(x)=x^2+3x+5$?
Answer (1)
To find the y -intercept, set x = 0 in the function f ( x ) = x 2 + 3 x + 5 .
Calculate f ( 0 ) = ( 0 ) 2 + 3 ( 0 ) + 5 = 5 .
The y -intercept is the point where the graph intersects the y -axis, which is ( 0 , f ( 0 )) .
Therefore, the y -intercept is ( 0 , 5 ) .
Explanation
Understanding the Problem We are given the function f ( x ) = x 2 + 3 x + 5 and asked to find the y -intercept of its graph. The y -intercept is the point where the graph of the function intersects the y -axis. This occurs when x = 0 .
Finding the y-intercept To find the y -intercept, we need to evaluate the function at x = 0 . That is, we need to find f ( 0 ) .
Calculating f(0) We substitute x = 0 into the function: f ( 0 ) = ( 0 ) 2 + 3 ( 0 ) + 5 f ( 0 ) = 0 + 0 + 5 f ( 0 ) = 5 Thus, the y -intercept is the point ( 0 , 5 ) .
Final Answer The y -intercept of the graph of the function f ( x ) = x 2 + 3 x + 5 is ( 0 , 5 ) .
Examples
Understanding the y-intercept of a function is crucial in many real-world applications. For example, if you are tracking the height of a ball thrown into the air with the function h ( t ) = − 5 t 2 + 20 t + 1 , where h ( t ) is the height in meters and t is the time in seconds, the y-intercept (h(0) = 1) tells you the initial height of the ball when it was thrown, which is 1 meter above the ground. Similarly, in business, if you have a cost function C ( x ) = 2 x 2 + 5 x + 100 , where C ( x ) is the cost of producing x items, the y-intercept (C(0) = 100) represents the fixed costs, which are the costs you have to pay even if you don't produce any items.
