Graph the function.
$f(x)=\sqrt{x}+3$
Answer (1)
The function f ( x ) = x + 3 is a vertical translation of the basic square root function y = x .
The graph of f ( x ) starts at (0, 3) and increases as x increases.
Key points on the graph include (1, 4), (4, 5), and (9, 6).
The graph is only defined for x ≥ 0 .
Explanation
Understanding the Function We are asked to graph the function f ( x ) = x + 3 . This is a transformation of the basic square root function.
The Basic Square Root Function The basic square root function y = x starts at the point (0, 0) and increases as x increases. The domain of the function is x ≥ 0 because we cannot take the square root of a negative number and get a real number.
Vertical Translation The function f ( x ) = x + 3 is a vertical translation of the basic square root function y = x . This means we shift the graph of y = x upwards by 3 units.
Plotting Points To graph f ( x ) = x + 3 , we can plot some points. When x = 0 , f ( 0 ) = 0 + 3 = 0 + 3 = 3 . So the graph starts at the point (0, 3). When x = 1 , f ( 1 ) = 1 + 3 = 1 + 3 = 4 . So the point (1, 4) is on the graph. When x = 4 , f ( 4 ) = 4 + 3 = 2 + 3 = 5 . So the point (4, 5) is on the graph. When x = 9 , f ( 9 ) = 9 + 3 = 3 + 3 = 6 . So the point (9, 6) is on the graph.
The Graph The graph of f ( x ) = x + 3 starts at the point (0, 3) and increases as x increases. It passes through the points (1, 4), (4, 5), and (9, 6). The graph is only defined for x ≥ 0 .
Examples
Understanding transformations of functions is crucial in many real-world applications. For example, in physics, the height of an object thrown upwards can be modeled by a quadratic function. If we add a constant to this function, it represents a vertical shift, which could indicate the object was thrown from a higher initial position. Similarly, in economics, cost functions can be shifted to represent fixed costs. Graphing these functions helps visualize the impact of these changes.
