Graph the function:
$f(x)=3 \sqrt{x}-4$
To analyze the function \(f(x) = 3\sqrt{x} - 4\), we begin by identifying its domain and range. The square root function \(\sqrt{x}\) is defined only for non-negative values, so the domain of our function is \(x \geq 0\). As \(x\) increases from \(0\) to \(\infty\), the term \(\sqrt{x}\) increases from \(0\) to \(\infty\). Consequently, \(3\sqrt{x}\) also ranges from \(0\) to \(\infty\), and subtracting 4 shifts this range down. Thus, the range of the function is \([-4, \infty)\).
Next, we determine the starting point (also known as the endpoint of the curve) and the intercepts. The starting point occurs at the lower bound of the domain, \(x = 0\). Evaluating the function at this point gives \(f(0) = 3\sqrt{0} - 4 = -4\). Therefore, the starting point is \((0, -4)\). Since the y-intercept occurs when \(x = 0\), the y-intercept is also \((0, -4)\). To find the x-intercept, we set \(f(x) = 0\):
\[3\sqrt{x} - 4 = 0\]
\[3\sqrt{x} = 4\]
\[\sqrt{x} = \frac{4}{3}\]
\[x = \left(\frac{4}{3}\right)^2 = \frac{16}{9} \approx 1.78\]
So, the x-intercept is \((\frac{16}{9}, 0)\).
To understand the shape and steepness of the curve, we calculate a few additional points:
The function starts at \((0, -4)\) and passes through \((\frac{16}{9}, 0)\), \((1, -1)\), \((4, 2)\), and \((9, 5)\), curving upwards as \(x\) increases due to the nature of the square root function and the vertical stretch by a factor of 3.
To graph the function, plot the starting point \((0, -4)\) and the calculated points \((1, -1)\), \((\frac{16}{9}, 0)\), \((4, 2)\), and \((9, 5)\) on a coordinate plane. Connect these points with a smooth curve that starts at \((0, -4)\) and extends infinitely to the right, showing an increasing, concave-downward growth pattern. This graph represents the transformation of \(y = \sqrt{x}\) stretched vertically by 3 and shifted down by 4 units. The final graph is defined by \(\boxed{f(x) = 3\sqrt{x} - 4}\).
Understanding how to graph functions like \(f(x) = 3\sqrt{x} - 4\) is essential in various fields, such as engineering and physics. For example, if you are modeling the speed of an object sliding down a curved ramp or the relationship between the distance traveled and time under specific constraints, you often use radical functions. Being able to identify the starting point (the initial state) and the intercepts (where the object hits a certain position or time) allows for accurate analysis of these real-world systems. Mastery of these transformations helps visualize how changing parameters—like increasing the slope or shifting the baseline—affects the overall behavior of the system.