Problem 70 - Algebra and Trigonometry Real Mathematics, Real People

Question

Graphical Analysis (a) use a graphing utility to graph the equation, (b) use the graph to approximate any \(x\) -intercepts of the graph, (c) set \(y=0\) and solve the resulting equation, and (d) compare the result of part (c) with the \(x\) -intercepts of the graph. $$y=2 x-\sqrt{15-4 x}$$

Step-by-Step Solution

Answer

To solve the exercise, graph the function, visually approximate the x-intercepts of the graph. Then, setting \(y=0\), solve the equation algebraically to find the x-intercepts. Verify that both methods yield the same x-intercepts.

Step 1: Graph the Equation

To graph the equation \(y=2x-\sqrt{15-4x}\), use a graphing utility and plug in the equation. Observe the shape, direction, and any intercepts of the graph.

Step 2: Find the x-intercepts of the Graph

The x-intercepts are the points where the graph crosses the x-axis and can be found by checking where on the graph, the y-value equals to zero. This can be approximated visually by inspecting the graph.

Step 3: Solve the Equation

Set \(y=0\) in the original equation and solve for \(x\). This means solving the equation \(0=2x-\sqrt{15-4x}\) algebraically.

Step 4: Compare the Results

Compare the x-intercepts found visually (from the graph) and algebraically (from the equation). They should concur, as both methods should yield the same x-intercepts.

Key Concepts

X-Intercepts

X-intercepts are crucial points in graphing equations where the graph crosses the x-axis. This usually indicates that the function has a root, or in other words, a solution to the equation at that particular value of x when the y-value is zero. When you're given a function like \(y = 2x - \sqrt{15 - 4x}\), finding the x-intercepts involves setting the y-value to zero and solving for x algebraically. Visually, this corresponds to observing where the graph of the equation touches or crosses the x-axis.

To facilitate a comprehensive understanding, let's break it down:

• Algebraic Approach: Set \(y=0\) and solve the equation \(0 = 2x - \sqrt{15 - 4x}\).
• Graphical Approach: Using a graph, locate the point(s) where the curve intersects the x-axis. These points are your x-intercepts.

Both the algebraic and graphical approaches should lead you to the same x-intercepts, thereby reinforcing your understanding of the concept.

Graphing Utility

Graphing utilities are indispensable tools in visualizing algebraic equations. They are designed to convert complex algebraic expressions into a visual format that is often easier to interpret. When faced with the task of graphing \(y = 2x - \sqrt{15 - 4x}\), you would use a graphing utility to input the equation and produce its graph. Some benefits of these utilities include:

• Instant visualization of the shape and direction of the graph.
• Ability to identify intercepts, maxima, minima, and trends quickly.
• Improved accuracy in approximating solutions.

A graphing utility not only offers visual insights but can also assist with cross-verifying algebraic solutions, providing an excellent means for double-checking your results. Always ensure that your tool is correctly set up for the scales and axes to get the most accurate representation possible.

Algebraic Solutions

An algebraic solution delves into the mathematical process of solving equations using algebraic manipulation. For finding the x-intercepts of \(y = 2x - \sqrt{15 - 4x}\), we set \(y=0\) and rearrange the equation to isolate and solve for x. This type of solution is exact, meaning it doesn't rely on visual approximations, and thus, it's critical for verifying the accuracy of graphical solutions.

The step-by-step process may involve:

• Isolating the radical expression.
• Squaring both sides to remove the radical.
• Solving the resulting quadratic equation.

Through these steps, we obtain a precise value for the x-intercepts, which should align with the graphical analysis. It's essential to carefully execute each step to avoid any algebraic errors that could lead to incorrect solutions.

Graphical Analysis

Graphical analysis involves interpreting the features of a graph to gain insight into the behavior of a mathematical function. This visual approach complements algebra by offering an immediate sense of the function's shape, including its intercepts, asymptotes, and turning points. When you graph an equation like \(y = 2x - \sqrt{15 - 4x}\), you're engaging with the graphical analysis by identifying where the graph intersects the axes—these points are your x- and y-intercepts.

Graphical analysis is particularly useful for:

• Understanding the general behavior of the function over different intervals.
• Estimating solutions when exact algebraic methods are complex or not readily available.
• Analyzing the symmetry, periodicity, and other unique features of the graph.

By comparing the graphical and algebraic solutions, you enhance your understanding of the underlying mathematical principles and ensure consistency in your results.