Problem 2 - Algebra 2

Question

Graph each function. \(y=\sqrt{x}-2\)

Step-by-Step Solution

Answer

To graph \(y=\sqrt{x}-2\), understand that this is a downshifted square root function. Start by plotting the point (0,-2), then proceed to mark other points such as (1,-1), (4,0), (9,1), etc., afterwards, draw the curve passing through these points.

Step 1: Understand the Parent Function

The parent function in this case is \(y=\sqrt{x}\), which is the square root function. The graph of a square root function ordinarily starts at point (0,0) and curves upwards and to the right. Any positive value of \(x\) yields a positive y-value.

Step 2: Understand the Shift

The 'minus 2' in the function indicates a vertical shift downwards by 2 units. This means the entire graph of the square root function will be shifted down by two places. Instead of starting at the point (0,0), it will start at (0,-2).

Step 3: Plot the Relevant Points

Begin by plotting the new starting point at (0, -2). For simplicity, choose easy values for \(x\) such as 1, 4, 9 and plot the corresponding \(y\) values making sure to account for the downward shift. For instance, when \(x=1\), \(y=\sqrt{1} -2 = -1\), so plot (1,-1). Repeat for other points and then sketch the curve connecting the points.

Key Concepts

Parent Functions

A parent function is the simplest form of a function that defines a family of functions sharing common characteristics. For square root functions, the parent function is typically represented as \(y=\sqrt{x}\). When graphed, this function produces a curve that starts at the origin point (0,0) and extends upward and to the right along the x-axis.

Understanding the parent function is fundamental because it serves as a basis for transformations like shifts and stretches. In its basic form, each positive value of \(x\) is associated with a corresponding positive \(y\) value.
For example:

• If \(x = 1\), then \(y = \sqrt{1} = 1\).
• If \(x = 4\), then \(y = \sqrt{4} = 2\).
• If \(x = 9\), then \(y = \sqrt{9} = 3\).

Recognizing this pattern helps in understanding how variations or transformations affect the function.

Vertical Shift

Vertical shifts are transformations that move a graph up or down along the y-axis. In our function, \(y=\sqrt{x}-2\), the "-2" indicates a vertical shift downward by 2 units. This alteration means every point on the graph of the parent function \(y=\sqrt{x}\) is moved 2 units lower.

Consequently, while the shape of the function remains the same, its position changes. The starting point of \(y=\sqrt{x}\) is (0,0), but with the vertical shift, it becomes (0,-2).
More examples include:

• With \(x = 1\), the function becomes \(y = \sqrt{1} - 2 = -1\), placing the point at (1, -1).
• For \(x = 4\), the outcome is \(y = \sqrt{4} - 2 = 0\), resulting in placing the point at (4, 0).
• And for \(x = 9\), we get \(y = \sqrt{9} - 2 = 1\), therefore the point is at (9, 1).

This shift is crucial in accurately sketching the graph.

Plotting Points

The process of plotting points is essential for drawing any graph, especially when dealing with transformations such as shifts. For the graph of \(y=\sqrt{x}-2\), it’s useful to select specific values of \(x\) that make calculations simple and result in clear points needing to be plotted.

Start by plotting the new starting point, which is (0, -2). Then, select easy values for \(x\) such as 1, 4, and 9.
For these selected values:

• At \(x = 1\), the calculation is \(y = \sqrt{1} - 2 = -1\), hence plot the point (1,-1).
• At \(x = 4\), the function produces \(y = \sqrt{4} - 2 = 0\), so place the point (4,0).
• At \(x = 9\), the result is \(y = \sqrt{9} - 2 = 1\), leading to the point (9,1).

Once these points are plotted, draw a smooth curve through them, illustrating the progression of the function. Be sure to extend the curve slightly beyond the last point to show the ongoing nature of the graph. This methodical approach in plotting simplifies the visualization of the function.