Algebra & Graphs
Interpreting equations and visualizing data for Physics & Chemistry.
1. Quadratic Equations
The Formula
For an equation $ax^2 + bx + c = 0$, the roots are given by:
$$ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} $$
Useful Properties:
- Sum of roots ($\alpha + \beta$) = $-b/a$
- Product of roots ($\alpha \beta$) = $c/a$
Used in: Equilibrium constants, Projectile motion.
2. Understanding Graphs
A. Straight Line ($y = mx + c$)
The most common graph.
- m: Slope ($\tan \theta$). If $m > 0$, line goes up. If $m < 0$, line goes down.
- c: Y-intercept (where line cuts y-axis).
Example: Kinetic Energy vs Frequency ($K_{max} = h\nu - \Phi$)
Slope = $h$, Intercept = $-\Phi$.
Slope = $h$, Intercept = $-\Phi$.
B. Rectangular Hyperbola ($xy = c$ or $y \propto 1/x$)
Inverse relationship. As $x$ increases, $y$ decreases.
- Curve never touches axes.
- Example: Boyle's Law ($PV = \text{constant}$).
C. Parabola ($y \propto x^2$)
Quadratic relationship.
- $y = kx^2$: Symmetric about y-axis (U-shape). Example: Kinetic Energy vs Velocity ($KE = \frac{1}{2}mv^2$).
- $y^2 = kx$: Symmetric about x-axis.
D. Exponential ($y = e^{-x}$)
Exponential Decay.
- Value drops rapidly at first, then slows down, never reaching zero.
- Example: Radioactive Decay ($N = N_0 e^{-\lambda t}$), First Order Kinetics.
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