Graph the function.
$f(x)=\sqrt{x-4}$
Answer (1)
Determine the domain of the function: x ≥ 4 .
Find the x-intercept: x = 4 .
Calculate points on the graph: (5, 1), (8, 2), (13, 3), (20, 4).
Sketch the graph starting from (4, 0), increasing as x increases. The final answer is the graph of the function f ( x ) = x − 4 .
Explanation
Understanding the Function We are asked to graph the function f ( x ) = x − 4 . This is a square root function, which is a transformation of the basic square root function y = x .
Finding the Domain First, we need to determine the domain of the function. Since we cannot take the square root of a negative number, we must have x − 4 ≥ 0 . Solving for x , we get x ≥ 4 . So, the domain of the function is all real numbers greater than or equal to 4.
Finding the X-Intercept Next, let's find the x-intercept. To do this, we set f ( x ) = 0 and solve for x : x − 4 = 0 Squaring both sides, we get: x − 4 = 0 So, x = 4 . The x-intercept is the point (4, 0).
Calculating Points on the Graph Now, let's find a few more points on the graph. We can choose some x values greater than 4 and calculate the corresponding f ( x ) values:
If x = 5 , then f ( 5 ) = 5 − 4 = 1 = 1 . So, the point (5, 1) is on the graph. If x = 8 , then f ( 8 ) = 8 − 4 = 4 = 2 . So, the point (8, 2) is on the graph. If x = 13 , then f ( 13 ) = 13 − 4 = 9 = 3 . So, the point (13, 3) is on the graph. If x = 20 , then f ( 20 ) = 20 − 4 = 16 = 4 . So, the point (20, 4) is on the graph.
Sketching the Graph Now we can plot these points and sketch the graph. The graph starts at (4, 0) and increases as x increases. The graph is a transformation of y = x , shifted 4 units to the right.
Final Answer The graph of the function f ( x ) = x − 4 starts at the point (4, 0) and extends to the right, increasing as x increases.
Examples
Square root functions are used in various real-world applications, such as calculating the speed of an object falling from a certain height or determining the length of a side of a square given its area. For example, if you drop a ball from a building, the time it takes to hit the ground can be modeled using a square root function. Understanding these functions helps in predicting and analyzing such phenomena.
