Algebraic Foundations and Linear/Quadratic Functions: Practice Exam Study Guide

Algebraic Expressions and Exponents

Simplifying Expressions and Eliminating Negative Exponents

Algebraic expressions often involve exponents, fractions, and roots. Simplifying such expressions requires applying the laws of exponents and algebraic manipulation.

• Law of Exponents: For any real numbers a and b, and integers m and n:

  • $a^m \cdot a^n = a^{m+n}$

  • $(a^m)^n = a^{mn}$

  • $\frac{a^m}{a^n} = a^{m-n}$

  • $a^{-n} = \frac{1}{a^n}$

• Example: Simplify $\left(\frac{a}{b^4}\right)^5 \left(\frac{b}{3a}\right)^2$

  • Apply the exponent to each term inside the parentheses.

  • Combine like terms and eliminate negative exponents.

Combining and Simplifying Rational Expressions

Rational expressions are fractions where the numerator and/or denominator are polynomials. Simplification involves finding common denominators and reducing the expression.

• Example: $\frac{2 - \frac{y}{x}}{2 + \frac{y}{x}}$

  • Find a common denominator for the terms in the numerator and denominator.

  • Simplify the resulting expression.

Radicals and Rationalization

Rationalizing Denominators and Numerators

Rationalizing involves removing radicals from the denominator or numerator of a fraction. This is often done by multiplying by a conjugate or an appropriate form of 1.

• Rationalizing the Denominator: To rationalize $\frac{1}{\sqrt{x} - \sqrt{y}}$, multiply numerator and denominator by the conjugate $\sqrt{x} + \sqrt{y}$.

• Rationalizing the Numerator: For $\frac{\sqrt{5+h} - \sqrt{2}}{h+3}$, multiply numerator and denominator by $\sqrt{5+h} + \sqrt{2}$.

• Example:

  • Rationalize $\frac{\sqrt{a} - 3}{a - 9}$ by multiplying by the conjugate $\sqrt{a} + 3$.

Linear Equations and Functions

Point-Slope and Slope-Intercept Forms

Linear equations can be written in several forms, most commonly point-slope and slope-intercept forms.

• Point-Slope Form: $y - y_1 = m(x - x_1)$, where $(x_1, y_1)$ is a point on the line and $m$ is the slope.

• Slope-Intercept Form: $y = mx + b$, where $m$ is the slope and $b$ is the y-intercept.

• Example: The equation of a line through $(2, 5)$ with slope $-2$ is $y - 5 = -2(x - 2)$.

• Finding the Equation from Two Points: Use the slope formula $m = \frac{y_2 - y_1}{x_2 - x_1}$, then substitute into point-slope form.

• Finding Slope and Intercept: Rearrange $2x - 5y = 12$ to $y = \frac{2}{5}x - \frac{12}{5}$ to identify slope and y-intercept.

Linear Growth and Decay Models

Modeling with Linear Functions

Linear models describe situations where a quantity increases or decreases at a constant rate.

• General Linear Model: $A = A_0 + rt$, where $A_0$ is the initial amount, $r$ is the rate of change, and $t$ is time.

• Example:

  • Increasing: $A = 18 + 0.3t$

  • Decreasing: $A = 18 - 0.3t$

Quadratic Functions

Standard Form, Vertex, and Intercepts

Quadratic functions have the form $f(x) = ax^2 + bx + c$. Their graphs are parabolas, and key features include the vertex, y-intercept, and x-intercepts.

• Vertex: The vertex of $f(x) = ax^2 + bx + c$ is at $x = -\frac{b}{2a}$.

• Y-Intercept: The y-intercept is $f(0) = c$.

• X-Intercepts: Solve $ax^2 + bx + c = 0$ using the quadratic formula:

  • $x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$

• Graph: The parabola opens upward if $a > 0$ and downward if $a < 0$.

• Example: For $f(x) = x^2 + 6x + 8$:

  • Vertex: $x = -\frac{6}{2} = -3$

  • Y-intercept: $8$

  • X-intercepts: Solve $x^2 + 6x + 8 = 0$

Factoring Quadratic Functions

Factoring expresses a quadratic as a product of two binomials, which is useful for finding x-intercepts.

• General Form: $ax^2 + bx + c = (mx + n)(px + q)$

• Example: $x^2 - 9x + 20 = (x - 4)(x - 5)$

Table: Quadratic Function Properties

Function	Vertex	Y-Intercept	X-Intercepts	Opens
$f(x) = x^2 + 6x + 8$	$(-3, -1)$	$8$	$x = -4, -2$	Up
$g(x) = -4x^2 + 4x + 3$	$(0.5, 4)$	$3$	Use quadratic formula	Down
$m(x) = 2x^2 - 4x + 1$	$(1, -1)$	$1$	Use quadratic formula	Up

Additional info: Table entries for x-intercepts of $g(x)$ and $m(x)$ require solving the quadratic formula explicitly.