Pre-Calculus 20 Summary

Pre-Calculus 20 Summary

Chapter 1: Sequences and Series

• Arithmetic Sequence

  • General term formula: $t<em>n = t</em>1 + (n - 1)d$

  • where:

    • $t_n$ = nth term

    • $t_1$ = first term

    • $n$ = number of terms

    • $d$ = common difference, calculated as $t<em>2 - t</em>1$

• Geometric Sequence

  • General term formula: $t<em>n = t</em>1(r)^{n-1}$

  • where:

    • $r$ = common ratio, calculated as $\frac{t<em>2}{t</em>1}$

• Sum of a Finite Series

  • Arithmetic series sum: $S<em>n = \frac{n(t</em>1 + t_n)}{2}$

  • Geometric series sum:

  • if $r <br>eq 1$: $S<em>n = \frac{t</em>1(r^n - 1)}{r - 1}$

  • if $r = 1$: $S<em>n = nt</em>1$

• Sum of an Infinite Series

  • The formula is applicable only for geometric sequences where |r| < 1:
    $S<em> = \frac{t</em>1}{1 - r}$

Chapter 2: Trigonometry

• Basic Trigonometric Ratios:

  • $\sin(\theta) = \frac{\text{opposite}}{\text{hypotenuse}}$

  • $\cos(\theta) = \frac{\text{adjacent}}{\text{hypotenuse}}$

  • $\tan(\theta) = \frac{\text{opposite}}{\text{adjacent}}$

• SOH CAH TOA

  • Mnemonic for remembering sin, cos, and tan.

• Law of Sines (Applicable for AAS, ASA, or SSA)

  • $\frac{a}{\sin A} = \frac{b}{\sin B} = \frac{c}{\sin C}$

• Law of Cosines (Applicable for SAS or SSS)

  • $c^2 = a^2 + b^2 - 2ab \cos C$

  • Rearranged forms allow calculation of angles or sides based on known values.

• Unit Circle:

  • Notable angles and corresponding coordinates are:

  • 0°: (1, 0)

  • 90°: (0, 1)

  • 180°: (-1, 0)

  • 270°: (0, -1)

Chapter 3: Quadratic Functions

• General Form:

  • $f(x) = a(x - p)^2 + q$

  • Alternate form: $f(x) = ax^2 + bx + c$

  • where:

    • (p, q) = vertex of the parabola

    • Axis of symmetry: $x = p$ (vertical line)

• Concavity:

  • If a > 0, the graph opens upwards.

  • If a < 0, the graph opens downwards.

• Width of Parabola:

  • Standard width if $|a| = 1$.

  • Narrow if |a| > 1.

  • Wide if |a| < 1 (fraction).

• Domain:

  • $D = {x | x ∈ ℝ}$

• Range:

  • Minimum if a > 0: $R = {y | y ≥ q}$

  • Maximum if a < 0: $R = {y | y ≤ q}$

• Intercepts:

  • x-intercepts (zeros): Set $y = 0$ and solve for x.

  • y-intercept: Set $x = 0$ and solve for y.

• Completing the Square: Steps:

  1. Factor out the coefficient of $x^2$ from the first two terms.

  2. Add $\left(\frac{\text{coefficient of } x}{2}\right)^2$ to the quadratic and subtract the same value, multiplied by the number in front of the bracket, from the constant term.

  3. Factor the resulting trinomial.

  4. Combine like terms.

• Graphing Using the Staircase Method:

  • Begin at the vertex.

  • Move 1 unit to the left/right, then move up/down by 1a, 3a, 5a as you progress from vertex.

Chapter 4: Quadratic Equations

• Solving Methods:

  • Graphing: Locate where the graph intersects the x-axis (x-intercepts).

  • Factoring: Factor the equation, set factors equal to zero, solve for x.

  • Completing the Square: Follow the steps from Chapter 3.

  • Quadratic Formula:
    $x = \frac{-b ± \sqrt{b^2 - 4ac}}{2a}$

  • From $ax^2 + bx + c = 0$

• Nature of Roots via the Discriminant:

  • Discriminant: $D = b^2 - 4ac$

  • If D > 0: Two real roots.

  • If $D = 0$: One real root.

  • If D < 0: No real roots.

Chapter 5: Radical Expressions and Equations

• Simplifying Radical Expressions:

  • Understand how to work with radicals based on integer powers.

• Solving Radical Equations: Steps:

  1. Isolate the square root portion of the equation.

  2. Square both sides of the equation (be attentive to binomials).

  3. Solve for the variable.

Chapter 6: Rational Expressions and Equations

• Non-Permissible Values: Values that make the denominator zero.

  • To find: Factor the denominator, set equal to zero, solve for the variable.

• Simplifying and Multiplying Rational Expressions:

  • Factor numerator and denominator, then cancel like terms.

• Dividing Rational Expressions:

  • Invert the second expression and multiply.

• Adding and Subtracting Rational Expressions:

1. Factor and simplify if possible.

   2. Find a common denominator.

   3. Adjust numerators accordingly, combining over one denominator before simplifying.

• Solving Rational Equations:

  • Multiply through by the common denominator to eliminate fractions, then solve for the variable.

Chapter 7: Absolute Value and Reciprocal Functions

• Evaluating Absolute Value Expressions:

  • $|a| = a$ and $|-a| = a$, where a is isolated and made positive.

• Graphing Absolute Value Functions:

  • All negative y-values flip to positive in the graph.

  • Domain: $D = {x | x ∈ ℝ}$

  • Range: $R = {y | y ≥ 0, y ∈ ℝ}$

• Piecewise Functions:

  • Linear and quadratic function handling; adjust according to the slope being positive or negative around the roots.

• Solving Absolute Value Equations:

  • Isolate the absolute value, applying $±$ to the equation side, separate it out to solve both resultant equations.

Chapter 8: Systems of Equations

• Solve by Graphing:

  • Identify intersections of two graphs, confirm through original equations.

• Solve by Substitution:

  1. Isolate a variable in one equation, substitute into the other.

  2. Solve for the second variable and backtrack for the first.

• Solve by Elimination:

  • Adjust one or both equations to eliminate a variable, then solve for the remaining variable.

Chapter 9: Linear and Quadratic Inequalities

• Graphing Inequalities:

  • Use solid lines for $≤, ≥$ and dashed for <, >.

  • Test points in the original equation to determine shading direction.

• Quadratic Inequalities:

  • Apply similar shading logic based on whether the quadratic opens upwards or downwards.

  • Factor and analyze intervals to ascertain where solutions hold true.