Problem 2 - Contemporary Precalculus

Question

In Exercises \(1-6,\) graph the equation by hand by plotting no more than six points and filling in the rest of the graph as best you can. Then use the calculator to graph the equation and compare the results. $$y=\sqrt{x+5}$$

Step-by-Step Solution

Answer

The coordinates of the six points are: \((-5, 0), (-4, 1), (-3, 1.41), (-2, 1.73), (-1, 2), (0, 2.24)\).

Step 1: Choose the x values

To start, choose six x values to obtain the respective y values. It's a good idea to choose a range of x values, both negative and positive, around zero. For example: $$x=-5, -4, -3, -2, -1, 0$$

Step 2: Find the y values

Compute the corresponding y values using the given equation $$y=\sqrt{x+5}$$. For x = -5: $$y=\sqrt{-5+5} = \sqrt{0} = 0$$ For x = -4: $$y=\sqrt{-4+5} = \sqrt{1} = 1$$ For x = -3: $$y=\sqrt{-3+5} = \sqrt{2} \approx 1.41$$ For x = -2: $$y=\sqrt{-2+5} = \sqrt{3} \approx 1.73$$ For x = -1: $$y=\sqrt{-1+5} = \sqrt{4} = 2$$ For x = 0: $$y=\sqrt{0+5} = \sqrt{5} \approx 2.24$$ The points obtained are: \((-5, 0), (-4, 1), (-3, 1.41), (-2, 1.73), (-1, 2), (0, 2.24)\)

Step 3: Plot the points on the graph

Plot the obtained points on a graph: 1. Label the x-axis from -6 to 1 and the y-axis from 0 to 3. 2. Mark the points \((-5, 0), (-4, 1), (-3, 1.41), (-2, 1.73), (-1, 2), (0, 2.24)\) on the graph.

Step 4: Draw the graph

Connect the plotted points smoothly to create a graph representing the equation $$y=\sqrt{x+5}$$. Make sure that the curve is smooth and represents each point correctly.

Step 5: Graph using a calculator

Input the equation $$y=\sqrt{x+5}$$ in a graphing calculator and graph it. Compare the calculator's graph to the one drawn by hand. Both graphs should look very similar, and their shape should be that of a square root function shifted to the left horizontally.

Key Concepts

Understanding the Square Root Function

The square root function is a fundamental mathematical concept and often appears in various math problems, including graphing. It is represented by the square root symbol \(\sqrt{}\). In this exercise, we examine the equation \(y=\sqrt{x+5}\) which is a modified version of the basic square root function \(y = \sqrt{x}\).

Here are some aspects of the square root function to consider:

• The function is defined only for non-negative values of its input. That's because the square root of a negative number is not a real number.
• It passes through the origin \((0,0)\) in its most basic form, but can be shifted left, right, up, or down based on any transformations made to the function.
• The graph of a basic square root function is a curve that rises slowly, becoming less steep as \(x\) increases.

For the function \(y = \sqrt{x+5}\), the starting point or the "vertex" is at \((-5,0)\), due to the \(+5\) within the square root which shifts the graph to the left by 5 units.

Effective Plotting Points

When graphing functions, plotting points is a crucial step. It involves selecting a variety of \(x\) values, calculating the corresponding \(y\) values using the function, then marking these points on the graph.

In this exercise, we selected six \(x\) values: \(-5, -4, -3, -2, -1,\) and \(0\). These values provide a good range to illustrate the behavior of the function \(y=\sqrt{x+5}\), especially since they include the \(x\)-value where the function starts \((-5)\).

Here's how you can determine suitable points for graphing:

• Choose \(x\) values that give whole numbers or easily manageable radicals when plugged into the square root function.
• Ensure there's a mix of values that extend the graph accurately over the desired range.
• Compute \(y\) for each \(x\) and note down the coordinates.

This method allows for a representation of the function's shape before using more sophisticated tools.

The Role of Graphing Calculators

Graphing calculators are powerful tools that help visualize functions like \(y=\sqrt{x+5}\). They allow users to input equations directly and generate the graph automatically.

Using a graphing calculator offers several benefits:

• Accurate visual representation: Graphs produced are precise, eliminating human error that may occur with manual plotting.
• Immediate feedback: Allows quick identification of graph trends and transformations.
• Function transformation checks: By comparing manually plotted points with the graphing calculator's output, you can confirm the accuracy of your hand-drawn graphs.

While it's great to rely on technology, building the skill to graph by hand enhances understanding, as it requires thinking through each step of the function's behavior.

Exploring Function Transformations

Function transformations involve changing the position and shape of a graph without altering its fundamental characteristics. For the square root function \(y=\sqrt{x+5}\), we see a transformation known as horizontal translation.

Here's how transformations apply to our function:

• Horizontal shifts occur when you add or subtract numbers inside the function's argument. Here, \(x+5\) moves the function left by 5 units since we are essentially solving \(x+5=0\) to find the new origin (-5).
• Vertical shifts move the graph up or down, depending on whether constants are added or subtracted outside the square root.
• Reflections and stretches can also occur but are not present in this specific example.

Understanding these transformations is crucial as they dictate how the graph of the function is altered from its basic form. Transformations allow for tailored modeling of data or scenarios.