Consider two types of nonlinear equations. What unique quality does each possess and how does that quality cause the graph's unique shape? Name two unique examples of these shapes in real-world situations.
Try describing hyperbolic, and parabolic
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The two types of nonlinear equations are quadratic equations and exponential equations. These equations possess unique qualities that cause distinct shapes on their graphs.
1. Quadratic Equations:
Quadratic equations are characterized by having the highest power of the variable as 2 (e.g., y = ax^2 + bx + c). The unique quality of quadratic equations is that they form a parabolic shape on their graph. The graph of a quadratic equation is a U-shaped curve called a parabola. The vertex, or the highest or lowest point on the graph, depends on the coefficients a, b, and c of the equation. If a > 0, the parabola opens upward and has a minimum point. Conversely, if a < 0, the parabola opens downward and has a maximum point.
Real-world examples of parabolic shapes can be seen in the trajectory of a launched projectile such as a thrown ball or a fireworks display. The curved arc that these objects follow is due to the influence of gravity, which causes the parabolic shape. Another example is the shape of some satellite dishes, which are parabolic in order to focus incoming signals onto a receiver at the focal point.
2. Exponential Equations:
Exponential equations are equations in which the variable appears as an exponent (e.g., y = ab^x). The unique quality of exponential equations is that they exhibit rapid growth or decay. On the graph, exponential equations form an upward or downward curve that increases or decreases at an accelerating rate. The steepness of the curve depends on the value of the base (b) in the equation.
Real-world examples of exponential growth can be observed in population growth, where the number of individuals increases rapidly over time in certain conditions. Compound interest calculations also follow an exponential growth pattern, where the amount of money in an account increases exponentially based on the interest rate. On the other hand, exponential decay can be seen in the process of radioactive decay, where the number of atomic nuclei remaining in a substance decreases exponentially over time.