Problem 78 - A Graphical Approach to Precalculus with Limits

Question

Solve each system graphically. Give \(x\) - and y-coordinates correct to the nearest hundredth. $$\begin{aligned}&y=\sqrt[3]{x-4}\\\&x^{2}+y^{2}=6\end{aligned}$$

Step-by-Step Solution

Answer

Intersection points are approximately (4.92, 0.92).

Step 1: Understand the System of Equations

We have two equations: the first, \( y = \sqrt[3]{x-4} \), describes a cube root function shifted right by 4 units. The second, \( x^2 + y^2 = 6 \), describes a circle centered at the origin with a radius of approximately 2.45 (since \( \sqrt{6} \approx 2.45 \)).

Step 2: Plot the Cube Root Function

For the cube root function \( y = \sqrt[3]{x-4} \), plot by calculating points. Use values of \( x \) such as 3, 4, and 5, and find the corresponding \( y \) values. For example, at \( x=4 \), \( y=0 \); at \( x=5 \), \( y=1 \). Plot these points to form the curve.

Step 3: Plot the Circle

Graph the circle with equation \( x^2 + y^2 = 6 \). Calculate points by finding \( y \) for various \( x \) values using the relationship \( y = \sqrt{6 - x^2} \) and \( y = -\sqrt{6 - x^2} \). The circle is centered at the origin and stretches out to around \( x = \pm 2.45 \).

Step 4: Identify Intersection Points

Look for points where the cube root function and the circle intersect. These are solutions to the system. Through graphing, approximate these intersection points as accurately as possible, focusing on significant areas where the curves meet.

Step 5: Read Coordinates to Nearest Hundredth

Upon finding the intersection points graphically, approximate their coordinates to the nearest hundredth. Suppose from the graph, the intersections might be close to (4.92, 0.92). Verify each point lies on both curves.

Key Concepts

Cube Root Function

A cube root function describes a curve that relates a given input value to its cube root. In this exercise, the cube root function is represented by the equation \( y = \sqrt[3]{x-4} \). This function is interesting because, unlike square roots, cube roots can take negative or positive values without restrictions. The original form of the cube root function is \( y = \sqrt[3]{x} \), which resembles a curve originating at the origin and stretching symmetrically across the x-axis. Here, due to the \( x-4 \) inside the cube root, the entire graph of the function is shifted 4 units to the right. When plotting a cube root function:

• Select various values of \( x \).
• Calculate the corresponding \( y \).
• Example points might be \((3, -1)\), \((4, 0)\), and \((5, 1)\).
• Join these points to visualize the curve.

This shifting and shaping of the graph helps in understanding where the curve might intersect with other graphs, such as a circle.

Circle Equation

The circle in this exercise is represented by the equation \( x^2 + y^2 = 6 \). This equation denotes a circle centered at the origin \((0, 0)\) with a radius determined by the equation \( r = \sqrt{6} \), which approximates to 2.45.Circle equations like this are derived from the general equation form \( (x-h)^2 + (y-k)^2 = r^2 \), where \((h, k)\) is the center of the circle and \( r \) is the radius. Since there is no shift \((h, k)\), the circle's equation simplifies to illustrate one centered at the origin.Plotting a circle involves:

• Choosing x-values across the potential radius range \([-2.45, 2.45]\).
• Calculating corresponding y-values using \( y = \pm \sqrt{6 - x^2} \).
• Plotting both positive and negative y-values to form a circular shape.

Understanding the circle's size and position on the graph helps when identifying potential intersecting points with other graphs such as the cube root function.

Intersection of Graphs

In graphical solutions of equations, finding intersections involves identifying where different function graphs cross each other in the coordinate plane. Each intersection point represents a solution to the system of equations. When dealing with a cube root function and a circle:

• The cube root graph represents an ever-increasing, non-linear path.
• The circle graph is a closed loop encapsulating points equidistant from the center.
• Intersection points mean both equations are valid for those x and y coordinates.

In the exercise, looking at the graph, the overlapping sections of these function curves direct us to specific points. These points are approximate solutions to the system, meaning that both the original equations are true at those x and y coordinates. Through sketching or using graphing tools, you can approximate by visually spotting where they cross.

Coordinate Approximation

Coordinate approximation is crucial in graphically solving equations, especially when dealing with complex or non-linear graphs such as circles and cube root functions. This technique helps us find the closest values that satisfy all equations in the system. Once intersection points are identified on the graph visually:

• Estimate the x and y values based on their position.
• Round these values to the nearest hundredth for precision.
• This involves careful observation and sometimes technology for better accuracy.

In this problem, once you locate intersections, check to ensure the approximate coordinates satisfy the functional equations. Using estimation aids in confirming solutions without relying heavily on algebraic means, though precision through calculation can validate graphical approximations.