Name the property illustrated by each equation – Embark on a mathematical journey to uncover the properties illustrated by various equations, from linear to trigonometric. Each equation holds a unique characteristic, revealing insights into the world around us.
Explore the slope-intercept form of linear equations, the parabolic nature of quadratics, the exponential growth and decay patterns, the logarithmic relationships, and the periodic oscillations of trigonometric functions. Discover how these properties find applications in diverse fields, from engineering to biology.
Properties Illustrated by Linear Equations
Linear equations are of the form y = mx + b, where m is the slope and b is the y-intercept. The slope represents the rate of change of the dependent variable (y) with respect to the independent variable (x).
A positive slope indicates that y increases as x increases, while a negative slope indicates that y decreases as x increases. The y-intercept represents the value of y when x is 0.
Effects of m and b on the Graph
| Value of m | Value of b | Effect on Graph ||—|—|—|| m > 0 | b > 0 | Line slopes up and intersects the y-axis above the origin || m > 0 | b < 0 | Line slopes up and intersects the y-axis below the origin | | m < 0 | b > 0 | Line slopes down and intersects the y-axis above the origin || m < 0 | b < 0 | Line slopes down and intersects the y-axis below the origin |
Properties Illustrated by Quadratic Equations
Quadratic equations are of the form y = ax^2 + bx + c.
The coefficient a determines the shape of the parabola, while the coefficients b and c determine the position and vertex of the parabola. The discriminant, which is given by b^2
4ac, determines the number and type of solutions.
* If the discriminant is positive, the equation has two distinct real solutions.
- If the discriminant is zero, the equation has one real solution (a double root).
- If the discriminant is negative, the equation has no real solutions.
Applications of Quadratic Equations, Name the property illustrated by each equation
* Modeling projectile motion
- Finding the roots of a polynomial
- Solving optimization problems
Properties Illustrated by Exponential Equations
Exponential equations are of the form y = a^x, where a is a positive constant and x is the variable. Exponential equations represent exponential growth or decay.* If a > 1, the equation represents exponential growth, where y increases rapidly as x increases.-
If a < 1, the equation represents exponential decay, where y decreases rapidly as x increases.
Applications of Exponential Equations
* Modeling population growth
- Radioactive decay
- Compound interest
Properties Illustrated by Logarithmic Equations
Logarithmic equations are of the form log(a) = x, where a is a positive constant and x is the variable. Logarithmic equations are used to solve for unknown exponents.
Relationship between Base, Exponent, and Value
| Base (a) | Exponent (x) | Value (log(a)) ||—|—|—|| 10 | 1 | 1 || 10 | 2 | 2 || 10 | 3 | 3 || e | 1 | 1 || e | 2 | 2 || e | 3 | 3 |
Properties Illustrated by Trigonometric Equations: Name The Property Illustrated By Each Equation
Trigonometric equations are of the form sin(x) = y, where x is the variable and y is a constant. Trigonometric equations are used to model periodic phenomena.
Applications of Trigonometric Equations
* Modeling sound waves
- Calculating angles in triangles
- Solving problems in engineering and physics
FAQ Compilation
What is the slope-intercept form of a linear equation?
The slope-intercept form of a linear equation is y = mx + b, where m represents the slope and b represents the y-intercept.
How does the discriminant of a quadratic equation determine the number and type of solutions?
The discriminant, calculated as b^2 – 4ac, determines the number and type of solutions to a quadratic equation. A positive discriminant indicates two real solutions, a discriminant of zero indicates one real solution, and a negative discriminant indicates two complex solutions.
