Graph the function. $f(x)=\sqrt{x}-7$
Answer (1)
The function f ( x ) = x − 7 is a transformation of the basic square root function y = x .
The transformation is a vertical shift down by 7 units.
Key points of the basic square root function are transformed by subtracting 7 from their y-coordinates.
The graph starts at (0, -7) and increases as x increases, resembling the shape of the basic square root function. f ( x ) = x − 7
Explanation
Understanding the Function We want to graph the function $f(x) =
\sqrt{x} - 7$. This is a transformation of the basic square root function.
Understanding the Basic Square Root Function The basic square root function y = x starts at the point (0, 0) and increases as x increases. The function is only defined for x ≥ 0 since we cannot take the square root of a negative number and get a real number.
Understanding the Transformation The given function f ( x ) = x − 7 is a vertical translation of the basic square root function. Specifically, it is shifted down by 7 units. This means that every point on the basic square root function is moved down 7 units.
Applying the Transformation to Key Points To graph the function, we can take some key points from the basic square root function and apply the transformation. For example:
(0, 0) becomes (0, 0 - 7) = (0, -7)
(1, 1) becomes (1, 1 - 7) = (1, -6)
(4, 2) becomes (4, 2 - 7) = (4, -5)
(9, 3) becomes (9, 3 - 7) = (9, -4)
Plotting the Graph Now, we plot these points (0, -7), (1, -6), (4, -5), and (9, -4) on a coordinate plane and draw a smooth curve through them. The curve starts at (0, -7) and increases as x increases, resembling the shape of the basic square root function.
Final Answer The graph of f ( x ) = x − 7 is the square root function shifted 7 units down.
Examples
The square root function is used in many real-world applications, such as calculating the distance to the horizon, determining the period of a pendulum, and modeling the growth of certain populations. Understanding transformations of functions, like the vertical shift in this problem, allows us to adapt these models to different scenarios. For example, if we were modeling the height of a plant, subtracting 7 might represent the initial height of the plant before it starts growing, f ( x ) = x − 7 .
