Problem 100 - Algebra and Trigonometry Real Mathematics, Real People

Question

Find the \(x\) - and \(y\) -intercepts of the graph of the equation. Use a graphing utility to verify your results. $$y=2 x \sqrt{x+7}$$

Step-by-Step Solution

Answer

The x-intercept and y-intercept of the given function \(y=2x \sqrt{x+7}\) is at the origin, i.e., (0,0).

Step 1: Find the y-intercept

To find the y-intercept, set \(x=0\) in the equation. Substituting \(x=0\) in \(y=2x \sqrt{x+7}\) gives \(y=0\). So the y-intercept is at the origin (0,0)

Step 2: Find the x-intercepts

The x-intercepts are found by setting the equation to \(y=0\). Setting \(y=0\) gives \(0=2x \sqrt{ x+7}\). To solve for x, divide both sides by \(2\sqrt{x+7}\), which gives us x=0 as the function is undefined for negative values of x. Hence the x-intercept is also at the origin (0,0).

Step 3: Summary of Intercepts

In this case, both the x-intercept and the y-intercept are at the origin (0,0).

Key Concepts

Graphing Utility

When presented with an equation such as \(y=2 x \sqrt{x+7}\), a visual representation can not only validate our calculations but also provide a more intuitive understanding of the function's behavior. A graphing utility is an indispensable tool in this process.

Whether using a graphing calculator or software, such as Desmos or GeoGebra, it is essential to know how to input the given function. Using these tools, students can visually inspect the graph for intercepts, symmetry, asymptotes, and other unique characteristics. After plotting \(y=2 x \sqrt{x+7}\), a graphing utility will clearly show that the intercepts are indeed at the origin, as also confirmed by the algebraic solution.

X-intercept Calculation

Finding the x-intercepts of a function involves determining where the graph crosses the x-axis. For the equation \(y=2 x \sqrt{x+7}\), setting \(y=0\) and solving for \(x\) is the procedural step for this calculation.

The solution obtained might include multiple intercepts, depending on the equation. However, in this case, after simplifying, we're left with \(x=0\) as the only solution. It is worth noting that for radical equations, the domain must be considered since the radicand, in this case \(x+7\), cannot be negative in order to have real solutions.

Y-intercept Calculation

Identifying the y-intercept is a matter of finding the point at which the function crosses the y-axis. This is typically found by setting \(x=0\) in the equation. For our function, substituting \(x=0\) simplifies the equation to \(y=0\), revealing that the y-intercept lies at the origin (0,0).

It's essential to recognize that not all functions will have a y-intercept, especially if they contain variables in the denominator that could be undefined at \(x=0\). Nevertheless, for functions expressed in a radical form like the one in our exercise, the y-intercept computation is straightforward as long as there are no restrictions on the variable under the radical.

Solving Radical Equations

Solving radical equations, such as the one given \(y=2 x \sqrt{x+7}\), requires attention to the domain of the function and properly isolating the radical expression. If this equation was to be solved for y, setting \(y=0\), we would need to eliminate the radical by squaring both sides of the equation—only after isolating the radical term.

However, this often introduces extraneous solutions, so it is crucial to check each potential solution by substituting back into the original equation. In this specific case, since the radical was already isolated and upon setting \(y=0\), we found that \(x=0\) is the only solution, the process was greatly simplified, and no extraneous solutions were introduced.