Graph the function.
$f(x)=\sqrt{x}+3$
Answer (1)
Recognize that f ( x ) = x + 3 is a vertical translation of the function g ( x ) = x by 3 units upward.
Graph the base function g ( x ) = x , which starts at the origin (0,0) and increases as x increases.
Shift the graph of g ( x ) = x upward by 3 units to graph f ( x ) = x + 3 . The starting point is now (0,3).
Plot points like (1,4), (4,5), and (9,6) to confirm the shape and position, then draw a smooth curve. The final answer is the graph of f ( x ) = x + 3 .
Explanation
Understanding the Function We want to graph the function f ( x ) = x + 3 . This is a square root function with a vertical shift.
Identifying the Transformation The basic square root function g ( x ) = x starts at the origin (0,0) and increases as x increases. The function f ( x ) is a transformation of g ( x ) by a vertical shift upwards by 3 units.
Shifting the Graph To graph f ( x ) = x + 3 , we shift the graph of g ( x ) = x upward by 3 units. This means the starting point of the graph will now be (0,3) instead of (0,0).
Plotting Points Let's plot a few points to confirm the shape and position of the graph:
When x = 0 , f ( 0 ) = 0 + 3 = 0 + 3 = 3 , so the point (0,3) is on the graph.
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.
Drawing the Graph Draw a smooth curve through the plotted points, starting at (0,3) and increasing as x increases. The graph should resemble the shape of the square root function, but shifted upward by 3 units.
Final Answer The graph of f ( x ) = x + 3 is a square root function shifted vertically upwards by 3 units.
Examples
Square root functions are used in various real-world applications, such as calculating the speed of a vehicle based on its skid marks or modeling the growth of certain populations. Understanding transformations of these functions, like the vertical shift in this problem, allows us to adapt the basic model to fit different scenarios. For example, if we were modeling the height of a plant, the '+3' could represent an initial height before any growth occurs. This helps in making accurate predictions and informed decisions based on mathematical models.
