What is the $y$-intercept of the graph of the function $f(x)=x^2+3 x+5$?
A. $(0,-5)$
B. $(0,-3)$
C. $(0,3)$
D. $(0,5)$

The y -intercept occurs when x = 0 .
Substitute x = 0 into the function f ( x ) = x 2 + 3 x + 5 .
Calculate f ( 0 ) = ( 0 ) 2 + 3 ( 0 ) + 5 = 5 .
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 = 0 + 0 + 5 = 5.

Stating the y-intercept Therefore, the y -intercept is the point ( 0 , 5 ) .


Examples
Imagine you're tracking the height of a plant over time, where x represents the number of days and f ( x ) represents the height of the plant. The y -intercept tells you the initial height of the plant when you started tracking it (at day 0). In this case, the plant started at a height of 5 units. Understanding intercepts helps in various real-world scenarios, such as determining initial values, break-even points, or starting conditions in mathematical models.

Answered by GinnyAnswer | 2025-07-05