Problem 8 - A Graphical Approach to College Algebra

Question

Skills Match each equation in Column I with the appropriate description in Column II. Do not use a calculator. A. Circle; center \((3,-4) ;\) radius 5 B. Parabola; opens left C. Parabola; opens upward D. Circle; center \((-3,4)\); radius 5 E. Parabola; opens right F. Circle; center \((0,0) ;\) radius \(\sqrt{5}\) G. No points on its graph H. Parabola; opens downward $$x^{2}+y^{2}=-4$$

Step-by-Step Solution

Answer

The equation matches with G: No points on its graph.

Step 1: Identify Equation Type

The given equation is \(x^2 + y^2 = -4\). This equation is in the form that typically represents a circle, which is \(x^2 + y^2 = r^2\) where \(r\) is the radius. For a circle, both \(x^2\) and \(y^2\) must yield a positive radius squared.

Step 2: Determine if the Radius is Valid

The equation \(x^2 + y^2 = -4\) suggests \((r^2 = -4)\). Since \(r^2\) is negative, it cannot represent a real circle, as radius squared must be non-negative in real numbers.

Step 3: Match with Column II

Since \(x^2 + y^2 = -4\) suggests no real solutions exist (a negative on the right side), it corresponds best with G. No points on its graph, as there are no real values of \(x\) and \(y\) that satisfy this equation.

Key Concepts

Circle Equation

The equation of a circle in the coordinate plane is typically expressed in the form \(x^2 + y^2 = r^2\) if the circle is centered at the origin \((0,0)\). Here, \(r\) represents the radius of the circle. The equation can also be written as \((x - h)^2 + (y - k)^2 = r^2\) for a circle centered at \((h, k)\). This format is essential to understand as it gives us information about the circle's center and radius.
For example, if you encounter the equation \((x - 3)^2 + (y + 4)^2 = 25\), the center of the circle is \((3, -4)\) and the radius is 5 (since \(5^2 = 25\)).

• The equation reveals the structural symmetry of the circle.
• The center \((h, k)\) indicates where the circle's midpoint is located.
• The radius \(r\) is the constant distance from the center to any point on the circle's edge.

The equation \(x^2 + y^2 = -4\), as shown in the original exercise, is problematic because \(r^2\) is negative, suggesting there isn't a circle with real points.

Radius

The radius of a circle is a crucial component, representing the distance from the center to any point on its perimeter. In algebraic terms, if you have a circle with an equation \((x - h)^2 + (y - k)^2 = r^2\), \(r\) is derived as \(r = \sqrt{r^2}\). This implies the radius is always a non-negative value.
It's important to recognize that in any valid circle equation, \(r^2\) must be a non-negative number. A negative value, like in the equation \(x^2 + y^2 = -4\), implies no real radius exists, hence no circle in the real plane.

• A negative \(r^2\) indicates errors in forming the circle equation or possibly no real circle at all.
• Understanding the radius helps visualize the size of the circle.
• The radius also aids in understanding the spatial relationships in geometric problems involving circles.

In simple terms, any time you're working with geometry or algebra involving circles, always check that the radius makes sense by ensuring \(r^2\) is non-negative.

Parabola

Parabolas are the graphical representations of quadratic functions and play a significant role in algebra. The general equation for a parabola is \(y = ax^2 + bx + c\) when it opens vertically, or \(x = ay^2 + by + c\) when it opens horizontally. The direction in which a parabola opens is determined by the coefficient of the square term.

Vertical Parabolas

A parabola that opens upward or downward takes the form \(y = ax^2 + bx + c\).
- If \(a > 0\), the parabola opens upward.
- If \(a < 0\), it opens downward.

Horizontal Parabolas

For parabolas taking the form \(x = ay^2 + by + c\), the orientation differs:
- When \(a > 0\), the parabola opens to the right.
- When \(a < 0\), it opens to the left.

• A parabola's vertex is the highest or lowest point, depending on its direction.
• The axis of symmetry is the line that vertically or horizontally bisects the parabola.
• Being familiar with these orientations helps in sketching parabolas and solving problems involving their graphs.

Understanding these basics allows you to determine how a parabola behaves and assists in applications ranging from physics problems to construction and design.