Problem 3 - Algebra 2

Question

Graph each function. \(y=\sqrt{x}-4\)

Step-by-Step Solution

Answer

The graph of function \(y = \sqrt{x} - 4\) starts from point (0, -4), then touches x-axis at (4, 0) and continues upwards and to the right. It is the standard square root function, shifted down by 4 units.

Step 1: Identification of basic function type

The function provided, \(y = \sqrt{x} - 4\), is a transformed square root function. The basic square root function is \(y = \sqrt{x}\), which only starts to increase gradually from x=0 on the graph.

Step 2: Understanding transformations

The '-4' in the function is a vertical shift. It indicates that compared to the basic square root function, this function is shifted downward by '4' units. So Mapped points of the graph would be like (0, -4), (1, -3), (4, 0), etc.

Step 3: Plot the graph

First, draw a cartesian coordinate plane. Then, start plotting the points identified in the previous step. The point at x=0 now lies at y=-4. The graph should touch the x-axis thus at x=4, y=0 and increase upwards and to the right from there. Reflect this altered shape of the square root function on the graph.

Key Concepts

Function Transformation

Function transformation is a core principle in understanding how different variations of a basic function affect its graph. In our original problem, the base function is the square root function, denoted as \(y = \sqrt{x}\). Function transformations involve altering this base function to produce a related function with specific differences.

• When we add or subtract a number outside the square root (like the \(-4\) in \(y = \sqrt{x} - 4\)), this represents a vertical transformation, shifting the graph up or down.
• Transformations can also include stretching or compressing the graph horizontally or vertically, or reflecting it across an axis, although these are not present in this specific problem.

Learning about function transformations helps in easily determining how any modifications to a function equation will influence its graph. It shows that even a small numerical change can significantly impact the overall shape and position of the graph on a cartesian plane. This is essential for graphing accurately and understanding the relationship between equations and their graphical representations.

Graphing Techniques

Graphing techniques are a series of steps or tips that can aid someone in creating an accurate plot of function transformations. When graphing the square root function \(y = \sqrt{x} - 4\), a few useful techniques include:

• Identify Key Points: The square root function starts at \((0, 0)\) in its basic form. After transformation, this point will be at \((0, -4)\). Identifying key points before graphing makes visualizing the transformation easier.
• Use Mapped Points: Identify a few values of \(x\) that provide simple calculations with the square root (like 0, 1, 4) and map these to their new \(y\) values. For example, at \(x = 4\), the traditional form gives \(y = 2\), and with the transformation \(y = \sqrt{4} - 4 = 0\).
• Draw Smooth Curves: The square root function graph is smooth and gradual. Make sure to illustrate this by connecting plotted points with a gentle curve.

Mastering these techniques will make it much easier to visualize how the transformation impacts the function's graph, providing a clearer understanding of function behavior.

Vertical Shift

A vertical shift is a specific type of transformation that moves the graph of a function up or down on the coordinate plane. In the exercise \(y = \sqrt{x} - 4\), the graph of the basic square root function is shifted down by 4 units.

• This transformation is indicated by \(-4\), meaning each original \(y\)-value from the \(y = \sqrt{x}\) graph is reduced by 4. If \((x, y)\) is a point on the original graph, then \((x, y-4)\) will be on the transformed graph.
• Vertical shifts maintain the shape of the graph while relocating it vertically. Therefore, characteristics like domain and range are adjusted accordingly. The new range, for example, but begins at \(y = -4\) instead of \(y = 0\).

Understanding vertical shifts allows for accurate prediction of how any function alterations will appear on a graph. Not only does it illustrate function behavior, but it also explains how these changes interact with other elements of the function's behavior such as the direction and steepness of the curve.