Graph the function.
$f(x)=\sqrt{x+3}$
Answer (1)
Determine the domain of the function: x ≥ − 3 .
Find the x-intercept: x = − 3 .
Calculate additional points: ( − 2 , 1 ) , ( 1 , 2 ) , ( 6 , 3 ) .
Sketch the graph starting at ( − 3 , 0 ) , passing through the calculated points, and increasing as x increases. The final answer is the graph of the function f ( x ) = x + 3 .
Explanation
Understanding the Function We are asked to graph the function f ( x ) = x + 3 . This is a square root function, which means the expression inside the square root must be non-negative.
Finding the Domain To find the domain of the function, we need to solve the inequality x + 3 ≥ 0 . Subtracting 3 from both sides, we get x ≥ − 3 . So, the domain of the function is all real numbers greater than or equal to -3.
Finding the X-Intercept To find the x-intercept, we set f ( x ) = 0 and solve for x : x + 3 = 0 Squaring both sides, we get: x + 3 = 0 So, x = − 3 . This means the graph starts at the point ( − 3 , 0 ) .
Finding Additional Points Now, let's find a few more points to help us sketch the graph. We'll choose some x values greater than or equal to -3.
Calculating f(-2) If x = − 2 , then f ( − 2 ) = − 2 + 3 = 1 = 1 . So, we have the point ( − 2 , 1 ) .
Calculating f(1) If x = 1 , then f ( 1 ) = 1 + 3 = 4 = 2 . So, we have the point ( 1 , 2 ) .
Calculating f(6) If x = 6 , then f ( 6 ) = 6 + 3 = 9 = 3 . So, we have the point ( 6 , 3 ) .
Sketching the Graph Now we have the points ( − 3 , 0 ) , ( − 2 , 1 ) , ( 1 , 2 ) , and ( 6 , 3 ) . We can plot these points and connect them with a smooth curve to sketch the graph of the function. The graph starts at ( − 3 , 0 ) and increases as x increases.
Final Answer The graph of f ( x ) = x + 3 starts at the point ( − 3 , 0 ) and increases as x increases. It passes through the points ( − 2 , 1 ) , ( 1 , 2 ) , and ( 6 , 3 ) .
Examples
Square root functions are used in various fields, such as physics and engineering, to model phenomena involving growth or decay rates that are proportional to the square root of a variable. For instance, the velocity of an object falling through a fluid might be modeled using a square root function, where the velocity depends on the square root of the distance fallen. Understanding and graphing these functions helps in predicting and analyzing such real-world scenarios.
