Algebra I Study Sheet Notes

ARITHMETIC PROPERTIES

• Commutative Property

  • Addition: $a + b = b + a$

  • Multiplication: $a \cdot b = b \cdot a$

• Associative Property

  • Addition: $a + (b + c) = (a + b) + c$

  • Multiplication: $a \cdot (b \cdot c) = (a \cdot b) \cdot c$

• Distributive Property: $a(b + c) = ab + ac$

• Identity Property

  • Addition: $a + 0 = a$

  • Multiplication: $a \cdot 1 = a$

• Inverse Property

  • Addition: $a + (-a) = 0$

  • Multiplication: $a \cdot (\frac{1}{a}) = 1$

• Reflexive Property: $a = a$

• Symmetric Property: If $a = b$, then $b = a$

• Transitive Property: If $a = b$ and $b = c$, then $a = c$

PROPERTIES OF EQUALITY

• Addition Property of Equality: If $a = b$, then $a + c = b + c$

• Subtraction Property of Equality: If $a = b$, then $a - c = b - c$

• Multiplication Property of Equality: If $a = b$, then $ac = bc$

• Division Property of Equality: If $a = b$, then $\frac{a}{c} = \frac{b}{c}$

• These properties also apply to inequalities, with one crucial exception: when multiplying or dividing by a negative number, you must flip the inequality sign.

RADICALS

• Square Root: Need two of a kind to move outside the radical.

  • Example: $\sqrt{24x^4y^3} = \sqrt{2 \cdot 2 \cdot 2 \cdot 3 \cdot x \cdot x \cdot x \cdot x \cdot y \cdot y \cdot y} = 2 \cdot x \cdot x \cdot y \sqrt{6y} = 2x^2y\sqrt{6y}$

• Cube Root: Need three of a kind to move outside the radical.

  • Example: $\sqrt[3]{24x^2y^3} = \sqrt[3]{2 \cdot 2 \cdot 2 \cdot 3 \cdot x \cdot x \cdot y \cdot y \cdot y} = 2 \cdot y \sqrt[3]{3x^2} = 2y\sqrt[3]{3x^2}$

FUNCTIONS PART 1

• Domain: The set of all possible x-values (inputs, independent variable).

• Range: The set of all possible y-values (outputs, dependent variable).

• Zeros: Also known as solutions, roots, or x-intercepts. They are the opposite of the binomial factor.

  • Example: $(x + 5)(x - 2)$. Zeros are -5 and 2.

• Function: A relation where each x-value has only one y-value.

• Not a Function: When the same x-value(s) have different range values.

• Vertical Line Test: If a vertical line crosses the graph at only one point, it is a function. If it crosses at more than one point, it is not a function.

FUNCTIONS PART 2

• Given $f(x) = -2x$ and $g(x) = x^2 + 1$

  • Find $f(2)$: $f(2) = -2(2) = -4$

  • Find $g(-2)$: $g(-2) = (-2)^2 + 1 = 4 + 1 = 5$

  • Find $f(2) + g(-2)$: $-4 + 5 = 1$

  • Find $3 \cdot f(2) - g(-2)$: $3 \cdot (-4) - 5 = -12 - 5 = -17$

THE DATA CYCLE

• Univariate Data: Data collected around a single characteristic.

• Bivariate Data: Data collected from two variables being compared.

• Bias: Occurs when the sampling is not completely random.

METHODS FOR COLLECTING DATA

• Observation: Watching and noting things as they happen.

• Survey: Asking people questions to gather information.

• Measurement: Using tools to find out how much, how long, or how heavy something is.

• Experiments: Conducting tests in a controlled way to get data.

SLOPE

• Slope ($m$) represents the rate of change (rise over run):
  $m = \frac{rise}{run} = \frac{y<em>2 - y</em>1}{x<em>2 - x</em>1}$

• Types of Slopes

  • Positive: Line goes up from left to right.

  • Negative: Line goes down from left to right.

  • Zero: Horizontal line.

  • Undefined: Vertical line.

• Slope Triangle: Rise over run representation on a graph.

EQUATIONS OF A LINE

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

• You can always find $b$ if given a point $(x, y)$ and the slope $m$.

• Parallel Lines: Have the same slope but different y-intercepts.

• Perpendicular Lines: Have negative (opposite) reciprocal slopes.

• HOY VUX

  • HOY: Horizontal line, 0 (zero) slope, equation is y = #

  • VUX: Vertical line, Undefined slope, equation is x = #

RULES FOR EXPONENTS

• Zero Exponent: Any non-zero number raised to the power of 0 is 1.

  • Example: $6^0 = 1$, $(xy)^0 = 1$, but $xy^0 = x \cdot 1 = x$

• Negative Exponent: Reciprocate the base and make the exponent positive.

  • Example:
    $\frac{-2yz}{(\frac{-2xy^{-2}z^3}{x^2})} = \frac{x^2}{y}$

• Multiplying Exponents: Multiply coefficients and add exponents.

  • Example: $5x(2x^2) = 10x^3$

• Dividing Exponents: Divide coefficients and subtract exponents.

  • Example: $\frac{8xy^{-1}}{4x^2y^{-3}} = 2x^{-1}y^{2} = \frac{2y^2}{x}$

• Power of a Power: Multiply exponents.

  • Example: $(2x^3)^2 = 2^2x^6 = 4x^6$

FACTORING PART 1

1. Find the Greatest Common Factor (GCF) first! Check both coefficients and variables.

2. Write in the form: $(GCF)(\text{remaining terms})$

   • Example: $5x^2y + 10xy^2 = 5xy(x + 2y)$

FACTORING PART 2

• Factoring Trinomials

  • Example: $x^2 + 4x - 5 = (x + 5)(x - 1)$

• Difference of Squares

  • Example: $x^2 - 9 = (x + 3)(x - 3)$

• Perfect Square Trinomial

  • Example: $x^2 - 6x + 9 = (x - 3)(x - 3) = (x - 3)^2$

• Factoring by grouping with a GCF
  $2x^2 + x - 6 = 0$
  $x = -12$
  $+ = 1$

• Zeros/Solutions/Roots/x-intercepts: Opposite of the binomial factor.

  • Example: $(x + 5)(x - 2) \implies \text{zeros: } -5 \text{ and } 2$

  • Example: $(2x - 3)(x + 2) \implies \text{zeros: } \frac{3}{2} \text{ and } -2$

QUADRATIC FUNCTIONS

• Standard Form: $f(x) = ax^2 + bx + c$

• If a > 0, the parabola faces up; if a < 0, the parabola faces down.

• Vertex: The point where the curve changes direction.

  • The x-coordinate of the vertex is given by: $x = \frac{-b}{2a}$

• Minimum: The lowest point on a parabola that opens upwards.

• Maximum: The highest point on a parabola that opens downwards.

QUADRATIC FORMULA

• Used to find the zeros of a quadratic equation.

• The quadratic equation must be in the form $ax^2 + bx + c = 0$

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

  • Example:
    $x^2 + 8x - 1 = 0$
    $a = 1, b = 8, c = -1$

    $x = \frac{-8 \pm \sqrt{8^2 - 4(1)(-1)}}{2(1)} = \frac{-8 \pm \sqrt{68}}{2} = \frac{-8 \pm 2\sqrt{17}}{2} = -4 \pm \sqrt{17}$

SYSTEMS OF EQUATIONS

• Solving by Graphing: Find the point where the lines intersect.

  • One Solution: Lines intersect.

  • No Solution: Lines are parallel (same slope, different y-intercepts).

  • Infinite Solutions: Lines are the same (same slope, same y-intercepts).

• Solving by Adding or Subtracting: Eliminate one variable by adding or subtracting the equations. Then, plug the result into one of the original equations to find the other variable. This method is useful when both equations are in standard form.
  $\begin{cases}2x + 3y = 6 \ 3x + 3y = 8\end{cases}$

• Solving by Substituting: Solve one equation for one variable (e.g., $y = \text{expression}$ or $x = \text{expression}$) and substitute that expression into the other equation.

GRAPHING INEQUALITIES

1. Solve the inequality for y.

2. Plot the y-intercept.

3. Use the slope to plot more points and draw the line.

   • Dashed line for > or <

   • Solid line for $\geq$ or $\leq$

4. Shade the region.

   • Shade above the line for > or $\geq$

   • Shade below the line for < or $\leq$

SYSTEMS OF INEQUALITIES

• Graph each inequality as described above.

• The overlapping shaded region contains the solutions to the system.

• Points on boundary lines adjacent to the shaded region are solutions if the lines are solid.

• The intersection point is a solution only if both lines are solid.

REGRESSIONS

• Use a graphing calculator (like DESMOS) to find the equation that best fits the data.

• Linear Regression: In DESMOS, use the form $y<em>1 \sim mx</em>1 + b$ to find the values of $m$ and $b$.

• Quadratic Regression: In DESMOS, use the form $y<em>1 \sim ax</em>1^2 + bx_1 + c$ to find the values of $a$, $b$, and $c$.

• Correlation Coefficient (r): $-1 \leq r \leq 1$. The closer $r$ is to -1 or 1, the stronger the correlation; the closer to 0, the weaker the correlation.

• Predictions: Use the equation from DESMOS, substitute the given value for $x$ or $y$, and solve for the other variable. You can also use DESMOS to verify your answer.

Term 1: Commutative Property of Addition
Definition 1: $a + b = b + a$
Term 2: Commutative Property of Multiplication
Definition 2: $a \cdot b = b \cdot a$
Term 3: Associative Property of Addition
Definition 3: $a + (b + c) = (a + b) + c$
Term 4: Associative Property of Multiplication
Definition 4: $a \cdot (b \cdot c) = (a \cdot b) \cdot c$
Term 5: Distributive Property
Definition 5: $a(b + c) = ab + ac$
Term 6: Identity Property of Addition
Definition 6: $a + 0 = a$
Term 7: Identity Property of Multiplication
Definition 7: $a \cdot 1 = a$
Term 8: Inverse Property of Addition
Definition 8: $a + (-a) = 0$
Term 9: Inverse Property of Multiplication
Definition 9: $a \cdot (\frac{1}{a}) = 1$
Term 10: Reflexive Property
Definition 10: $a = a$
Term 11: Symmetric Property
Definition 11: If $a = b$, then $b = a$
Term 12: Transitive Property
Definition 12: If $a = b$ and $b = c$, then $a = c$
Term 13: Addition Property of Equality
Definition 13: If $a = b$, then $a + c = b + c$
Term 14: Subtraction Property of Equality
Definition 14: If $a = b$, then $a - c = b - c$
Term 15: Multiplication Property of Equality
Definition 15: If $a = b$, then $ac = bc$
Term 16: Division Property of Equality
Definition 16: If $a = b$, then $\frac{a}{c} = \frac{b}{c}$
Term 17: Square Root
Definition 17: Need two of a kind to move outside the radical.
Term 18: Cube Root
Definition 18: Need three of a kind to move outside the radical.
Term 19: Domain
Definition 19: The set of all possible x-values (inputs, independent variable).
Term 20: Range
Definition 20: The set of all possible y-values (outputs, dependent variable).
Term 21: Zeros
Definition 21: Also known as solutions, roots, or x-intercepts. They are the opposite of the binomial factor.
Term 22: Function
Definition 22: A relation where each x-value has only one y-value.
Term 23: Not a Function
Definition 23: When the same x-value(s) have different range values.
Term 24: Vertical Line Test
Definition 24: If a vertical line crosses the graph at only one point, it is a function. If it crosses at more than one point, it is not a function.
Term 25: Univariate Data
Definition 25: Data collected around a single characteristic.
Term 26: Bivariate Data
Definition 26: Data collected from two variables being compared.
Term 27: Bias
Definition 27: Occurs when the sampling is not completely random.
Term 28: Observation
Definition 28: Watching and noting things as they happen.
Term 29: Survey
Definition 29: Asking people questions to gather information.
Term 30: Measurement
Definition 30: Using tools to find out how much, how long, or how heavy something is.
Term 31: Experiments
Definition 31: Conducting tests in a controlled way to get data.
Term 32: Slope ($m$)
Definition 32: Represents the rate of change (rise over run): $m = \frac{rise}{run} = \frac{y<em>2 - y</em>1}{x<em>2 - x</em>1}$
Term 33: Positive Slope
Definition 33: Line goes up from left to right.
Term 34: Negative Slope
Definition 34: Line goes down from left to right.
Term 35: Zero Slope
Definition 35: Horizontal line.
Term 36: Undefined Slope
Definition 36: Vertical line.
Term 37: Slope-Intercept Form
Definition 37: $y = mx + b$ where $m$ is the slope and $b$ is the y-intercept.
Term 38: Parallel Lines
Definition 38: Have the same slope but different y-intercepts.
Term 39: Perpendicular Lines
Definition 39: Have negative (opposite) reciprocal slopes.
Term 40: HOY
Definition 40: Horizontal line, 0 (zero) slope, equation is y = #
Term 41: VUX
Definition 41: Vertical line, Undefined slope, equation is x = #
Term 42: Zero Exponent
Definition 42: Any non-zero number raised to the power of 0 is 1.
Term 43: Negative Exponent
Definition 43: Reciprocate the base and make the exponent positive.
Term 44: Multiplying Exponents
Definition 44: Multiply coefficients and add exponents.
Term 45: Dividing Exponents
Definition 45: Divide coefficients and subtract exponents.
Term 46: Power of a Power
Definition 46: Multiply exponents.
Term 47: Find the Greatest Common Factor (GCF) first!
Definition 47: Check both coefficients and variables.
Term 48: Factoring Form
Definition 48: Write in the form: $(GCF)(\text{remaining terms})$
Term 49: Zeros/Solutions/Roots/x-intercepts
Definition 49: Opposite of the binomial factor.
Term 50: Standard Form of a Quadratic Function
Definition 50: $f(x) = ax^2 + bx + c$
Term 51: Quadratic Formula
Definition 51: Used to find the zeros of a quadratic equation.
Term 52: Quadratic Formula
Definition 52: $x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$
Term 53: Solving Systems of Equations by Graphing
Definition 53: Find the point where the lines intersect.
Term 54: One Solution in Systems of Equations
Definition 54: Lines intersect.
Term 55: No Solution in Systems of Equations
Definition 55: Lines are parallel (same slope, different y-intercepts).
Term 56: Infinite Solutions in Systems of Equations
Definition 56: Lines are the same (same slope, same y-intercepts).
Term 57: Solving Systems of Equations by Substituting
Definition 57: Solve one equation for one variable and substitute that expression into the other equation.
Term 58: Graphing Inequalities - Dashed Line
Definition 58: Dashed line for > or <
Term 59: Graphing Inequalities - Solid Line
Definition 59: Solid line for $\geq$ or $\leq$
Term 60: Graphing Inequalities - Shade Above
Definition 60: Shade above the line for > or $\geq$
Term 61: Graphing Inequalities - Shade Below
Definition 61: Shade below the line for < or $\leq$
Term 62: Systems of Inequalities - Solution Region
Definition 62: The overlapping shaded region contains the solutions to the system.
Term 63: Regressions
Definition 63: Use a graphing calculator (like DESMOS) to find the equation that best fits the data.
Term 64: undefined
Definition 64: In DESMOS