Algebra 1 Master Review

• Variable: A letter representing a number.
  Ex: In $3x + 5$, the variable is x.
• Coefficient: The number multiplied in front of a variable.
  Ex: In $\mathbf{5}x - 2$, the coefficient is 5.
• Constant: A number by itself (no variable).
  Ex: In $2x + \mathbf{7}$, the constant is 7.
• Leading Coefficient: The coefficient of the term with the highest exponent.
  Ex: $7 - 2x + 5x^3$. Highest power is 3. Leading Coeff is 5.
• Substitution: Plugging a number in for a variable.
  Ex: Evaluate $3x + 2$ when $x=4$.
  $3(4) + 2 = 12 + 2 =$ 14.

• Adding: Combine Like Terms only.
  $(2x + 5) + (3x - 2) \rightarrow (2x+3x) + (5-2) \rightarrow$ $5x + 3$
• Subtracting: Distribute the Negative Sign to the second group, then Add.
  $(4x + 7) - (x - 3) \rightarrow 4x + 7 \mathbf{-x + 3} \rightarrow$ $3x + 10$
• Multiplying (Distribute/FOIL): Multiply coefficients, ADD exponents.
  Ex: $3x(2x + 5) \rightarrow (3 \cdot 2)(x \cdot x) + (3 \cdot 5)(x) \rightarrow$ $6x^2 + 15x$
  Ex: $(x+2)(x+3) \rightarrow x^2 + 3x + 2x + 6 \rightarrow$ $x^2 + 5x + 6$
• Dividing: Divide coefficients, SUBTRACT exponents.
  Ex: $\frac{12x^5}{4x^2} \rightarrow (12 \div 4)x^{(5-2)} \rightarrow$ $3x^3$

• Zero Rule: Anything^0 = 1.
  $5^0 =$ 1
• Negative Rule: Negative exponent? Flip the fraction.
  $x^{-4} \rightarrow$ $\frac{1}{x^4}$
• Simplifying Radicals:
  Square Root: $\sqrt{75} = \sqrt{25 \cdot 3} \rightarrow$ $5\sqrt{3}$
  Cube Root: $\sqrt[3]{24} = \sqrt[3]{8 \cdot 3} \rightarrow$ $2\sqrt[3]{3}$

• Multistep Strategy:
  Example: $3(x - 2) + 4 = 19$
  1. Distribute: $3x - 6 + 4 = 19$
  2. Combine: $3x - 2 = 19$
  3. Add 2: $3x = 21$
  4. Divide by 3: $x = 7$
• Variables on Both Sides: Move smaller $x$ to larger $x$.
  Ex: $5x + 2 = 3x + 10$
  Subtract $3x$: $2x + 2 = 10 \rightarrow 2x = 8 \rightarrow$ $x = 4$
• Literal Equations: Treat extra letters like numbers.
  Ex: Solve $A = lw$ for $w$ $\rightarrow$ Divide by $l$ $\rightarrow$ $w = \frac{A}{l}$

• THE GOLDEN RULE: Multiply/Divide by Negative $\rightarrow$ FLIP SYMBOL.
  $-2x > 10 \rightarrow$ Divide by -2 $\rightarrow$ $x < -5$
• Visual Guide:

• Definition: X cannot repeat. Each Input has exactly ONE Output.
• Table Example:

  X	Y
  1	5
  2	8
  1	9

  X=1 repeats with different Y's.
  Answer: NOT A FUNCTION.
• Vertical Line Test: If a vertical line touches the graph more than once, it fails.

• Slope-Intercept Form: $y = mx + b$
  Ex: $y = -2x + 3 \rightarrow$ Slope $m=$ -2, Y-int $b=$ 3
• Point-Slope Form: $y - y_1 = m(x - x_1)$
  Ex: $y - 4 = 3(x - 2) \rightarrow$ Slope $m=$ 3, Point is $(2, 4)$
• Slope Formula: $m = \frac{y_2 - y_1}{x_2 - x_1}$
  $(1, 3)$ and $(5, 11)$.
  $m = \frac{11 - 3}{5 - 1} = \frac{8}{4} =$ 2
• HOY VUX (Special Lines):
  HOY: Horizontal line, 0 Slope, Equation is $\mathbf{Y} = \#$
  VUX: Vertical line, Undefined Slope, Equation is $\mathbf{X} = \#$

• Visual Solution: The point $(x,y)$ where two lines CROSS.
• Example:
  $y = 2x$ and $y = x + 1$
  Graph them. They cross at $(1, 2)$.
• Types of Solutions:

  One Solution	No Solution	Infinite Solutions
  Lines Cross Once $(x, y)$	Parallel Lines Same Slope Different Y-int	Same Line On top of each other

• 1. GCF (Greatest Common Factor): ALWAYS check this first!
  $4x^2 + 8x \rightarrow$ Pull out $4x \rightarrow$ $4x(x+2)$
• 2. Difference of Squares: Two terms subtracted. Both perfect squares.
  $x^2 - 49 \rightarrow$ $(x+7)(x-7)$
• 3. Trinomials ($x^2 + bx + c$): Multiplies to C, Adds to B.
  $x^2 + 7x + 12 \rightarrow$ (3 and 4) $\rightarrow$ $(x+3)(x+4)$
• 4. Slip & Slide ($ax^2 + bx + c$): When number in front $>1$.
  $2x^2 + 7x + 3$
  1. Multiply $a \cdot c$ ($2 \cdot 3 = 6$): $x^2 + 7x + 6$
  2. Factor: $(x+6)(x+1)$
  3. Divide by $a$ (2): $(x + \frac{6}{2})(x + \frac{1}{2})$
  4. Simplify/Slide: $(x+3)(2x+1)$

Parent: $f(x)$

• Vertical Shift ($+k$): Moves Up/Down.
  Ex: $x^2 + 3 \rightarrow$ Up 3
• Horizontal Shift ($x-h$): Inside is Opposite!
  Ex: $(x-2)^2 \rightarrow$ Right 2.
  Ex: $(x+5)^2 \rightarrow$ Left 5.
• Reflection ($-f(x)$): Negative in front flips it over X-axis.
• Dilation ($a \cdot f(x)$):
  If $a > 1$ (e.g., $3x^2$): Stretch (Skinny)
  If $0 < a < 1$ (e.g., $0.5x^2$): Compression (Wide)

• Mean: The Average. Add all numbers and divide by how many there are.
• Median: The Middle number. (Order least to greatest first).
• Mode: The number that appears the most.
• Range: The spread. Range = Max - Min.
• IQR (Interquartile Range): Middle 50%. Formula: $Q3 - Q1$.
  Ex: $Q3 = 20, Q1 = 15$
  $\rightarrow IQR = 20 - 15 = $ 5.
• Z-Score: Number of standard deviations a value is from the mean.
  Formula: $z = \frac{\text{Value} - \text{Mean}}{\text{Standard Deviation}}$
  Ex: Value=85, Mean=75, SD=5.
  $z = \frac{85-75}{5} = \frac{10}{5} =$ 2.
• Correlation ($r$): How close dots are to a line.
  $r \approx 1$ or $-1$: Strong Correlation.
  $r \approx 0$: Weak/No Correlation.