Precalculus Study Notes: Equations, Functions, and Applications

Equations and Inequalities

Solving Quadratic Equations

Quadratic equations are equations of the form ax2 + bx + c = 0. They can be solved using various methods, including factoring, completing the square, and the quadratic formula.

• Quadratic Formula: The solution to ax2 + bx + c = 0 is given by: $x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$

• Example: Solve 2x2 - 3x + 1 = 0 using the quadratic formula.

  • Here, a = 2, b = -3, c = 1.

  • $x = \frac{-(-3) \pm \sqrt{(-3)^2 - 4 \cdot 2 \cdot 1}}{2 \cdot 2}$

  • $x = \frac{3 \pm \sqrt{9 - 8}}{4}$

  • $x = \frac{3 \pm 1}{4}$

  • So, $x = 1$ or $x = 0.5$

Solving Radical Equations

Radical equations involve variables inside a root. To solve, isolate the radical and then square both sides.

• Example: Solve $\sqrt{2(x + 16)} - 4 = x$

• Isolate the radical: $\sqrt{2(x + 16)} = x + 4$

• Square both sides: $2(x + 16) = (x + 4)^2$

• Expand and solve the resulting quadratic equation.

Graphs and Functions

Line of Best Fit

A line of best fit is a straight line that best represents the data on a scatter plot. It can be found using a graphing utility or statistical methods.

• Equation of a Line of Best Fit: $y = mx + b$

• Example: Given data points, the line of best fit is found to be $y = 2.036x - 0.5211$

• This line can be used to predict values and analyze trends.

Vertex of a Parabola

The vertex of a parabola given by $f(x) = ax^2 + bx + c$ is the point $(h, k)$ where the function reaches its maximum or minimum value.

• Vertex Formula: $h = -\frac{b}{2a}$, $k = f(h)$

• Example: For $f(x) = -2x^2 + 4x - 9$, $a = -2$, $b = 4$ $h = -\frac{4}{2 \times -2} = 1$ $k = f(1) = -2(1)^2 + 4(1) - 9 = -2 + 4 - 9 = -7$ Vertex: (1, -7)

Polynomial and Rational Functions

Complex Zeros of Polynomials

Polynomials may have real or complex zeros. Complex zeros occur in conjugate pairs if the polynomial has real coefficients.

• Example: For $f(x) = x^3 - 8x^2 + 25x - 26$, the complex zeros are $2,$3+2i$, and $3 - 2i$.

• Example: For $f(x) = x^4 + 4x^3 + 4x + 36x - 45$, the zeros are $1 - 5i$, $3$, $0$, and $1 + 5i$.

Exponential and Logarithmic Functions

Solving Logarithmic Equations

Logarithmic equations can be solved by applying logarithmic properties and converting to exponential form if necessary.

• Example: Solve $\log(8x + 1) = 1 + \log(x - 8)$

• Use the property $\log a - \log b = \log \left(\frac{a}{b}\right)$

• Rewrite: $\log(8x + 1) - \log(x - 8) = 1$

• $\log\left(\frac{8x + 1}{x - 8}\right) = 1$

• Convert to exponential form: $\frac{8x + 1}{x - 8} = 10^1 = 10$

• Solve for $x$.

Exponential Growth Applications

Exponential growth is modeled by the equation $P = P_0 e^{kt}$, where $P_0$ is the initial amount, $k$ is the growth rate, and $t$ is time.

• Example (a): If $P_0 = 1000$, $P = 1800$ after 1 day, find $P$ after 3 days.

  • First, solve for $k$ using $1800 = 1000 e^{k \cdot 1}$

  • $e^k = 1.8 \implies k = \ln(1.8)$

  • After 3 days: $P = 1000 e^{3k} = 1000 (e^k)^3 = 1000 (1.8)^3 = 1000 \times 5.832 = 5832$

• Example (b): How long until $P = 70,000$?

  • $70,000 = 1000 e^{kt}$

  • $e^{kt} = 70$

  • $kt = \ln(70)$

  • $t = \frac{\ln(70)}{k}$

  • Substitute $k = \ln(1.8)$ to find $t$.

Systems of Equations and Inequalities

Solving Systems Using Row Operations

Systems of linear equations can be solved using matrices and row operations (Gaussian elimination).

• Example: Solve the system:

  • $14x + 7y = 22$

  • Write as an augmented matrix and use row operations to solve for $x$ and $y$.

Summary Table: Key Formulas and Concepts

Topic	Key Formula	Example
Quadratic Formula	$x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$	$2x^2 - 3x + 1 = 0$
Vertex of Parabola	$h = -\frac{b}{2a}$	$f(x) = -2x^2 + 4x - 9$
Exponential Growth	$P = P_0 e^{kt}$	$P_0 = 1000$, $P = 1800$ after 1 day
Logarithmic Equation	$\log a - \log b = \log \left(\frac{a}{b}\right)$	$\log(8x + 1) = 1 + \log(x - 8)$
System of Equations	Row operations on augmented matrix	$14x + 7y = 22$