Problem 67 - Algebra and Trigonometry

Question

In Exercises 63–68, find the solution set for each system by graphing both of the system’s equations in the same rectangular coordinate system and finding points of intersection. Check all solutions in both equations. $$ \left\\{\begin{array}{l} {x=(y+2)^{2}-1} \\ {(x-2)^{2}+(y+2)^{2}=1} \end{array}\right. $$

Step-by-Step Solution

Answer

The solution set of the system is the intersection points of the parabola and the circle on the graph. The exact coordinates of these points can be calculated by equating and solving the two equations.

Step 1: Graph the First Equation

The first equation can be rewritten in the format of parabola equation \((y - h)^2 = 4a(x - k)\), where \((h, k)\) is the vertex of the parabola. Here, the vertex is at \((-2, -1)\). Plot this point and shape of the parabola upwards.

Step 2: Graph the Second Equation

The second equation is in the standard form of a circle equation \((x - h)^2 + (y - k)^2 = r^2\), where \((h, k)\) is the center of the circle and \(r\) is the radius. In this case, the center of the circle is at \((2, -2)\) and the radius is \(1\). Plot this point and draw the circle with radius 1.

Step 3: Find Intersection Points

The intersection points of the parabola and the circle are the solution to the system. To find these points, plot where the parabola and the circle meet on the graph.

Step 4: Check Solutions

Substitute the coordinates of the intersection points into the original equations to verify they are indeed the solutions. For an exact solution, both equations would need to be solved for \(x\) and \(y\) and set equal to each other.

Key Concepts

Parabola Equation

A parabola is a symmetric, U-shaped curve that can be oriented vertically or horizontally. The general equation for a parabola that opens to the right or left is given by:\[(y - k)^2 = 4a(x - h)\]Here,

• \( (h, k) \) is the vertex of the parabola, or the "turning point" where the direction changes.
• \( a \) represents the distance from the vertex to the focus and dictates how "wide" or "narrow" the parabola appears.

In the context of the provided exercise, the parabola's equation is rewritten as \((y + 2)^2 = x + 1\). Let's break it down:

• The vertex of this parabola is at \[(-2, -1)\], which we deduced by rewriting the equation in a way that visualizes the vertex.
• This parabola will open horizontally to the right, because the equation has the form \((y-k)^2 \).

To graph this equation correctly, start by plotting the vertex at \((-2, -1)\) and sketch the curve, ensuring that it maintains a symmetrical shape across its axis.

Circle Equation

Circles in a coordinate plane are defined by the equation:\[(x - h)^2 + (y - k)^2 = r^2\]Let's interpret this equation:

• \( (h, k) \) is the center of the circle.
• \( r \) is the radius, which is the constant distance from the center to any point on the circle.

For the system of equations, the circle’s equation \((x - 2)^2 + (y + 2)^2 = 1\) features:

• The center at \((2, -2)\).
• A radius of \(1\), indicating the distance from the center to any point on the boundary of the circle.

To graph the circle, plot the center at \((2, -2)\) and draw a circle with all points exactly one unit away from the center. The circle should appear perfectly round as it maintains constant distance.

Intersection Points

Finding intersection points of graphs involves determining where two or more graphs meet. These points provide solutions to the system of equations.
For the given system, this means identifying where the parabola and circle intersect. To do this:

• Graph both shapes as outlined in their respective equations.
• Look for points where both graphs share the same coordinates.

Once you spot potential intersection points by eye, verify them algebraically by substituting the coordinates into both original equations to ensure they satisfy each equation. In the step-by-step process:

• Plot the parabola and circle on the same coordinate plane.
• Check where the shapes physically overlap.

By doing this, you can find the most accurate solutions. This step isn't just about visualization – verification of solutions is crucial for ensuring the graphs are correct and that calculations align with expectations.