Mathematical Facts, Constants & Formulas
Grade 8: Linear Functions (y = mx + b), Exponent Laws & The Pythagorean Theorem
~9 min read
HESTEN ACADEMIC REFERENCE ARCHIVE
CALL NO: QA107.H47 2026 • DEWEY: 510.71 • VOL. VIII
Volume VIII • Linear Foundations & Pre-Algebra Codex
Eighth Grade Mathematics Reference Codex
The Complete Scholastic Guide to Real Numbers, Exponent Laws, Scientific Notation, Linear Functions ($y=mx+b$), Systems of Equations, The Pythagorean Theorem, 3D Curved Volumes & Synoptic Concordance
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An authoritative academic handbook for eighth-grade scholars, secondary educators, and STEM specialists. Codifying the real number continuum, integer exponent mechanics, Cartesian slope functions, linear systems, the Euclidean Pythagorean metric, curved solid mensuration, and computational mental arithmetic.
§ 8.1
The Real Number Continuum: Rational vs. Irrational Numbers
DEFINITION 8.1.1
Topic: The Real Number System ($\mathbb{R}$)
- Rational Number ($\mathbb{Q}$) — Any number that can be expressed as a ratio of two integers $\frac{a}{b}$ ($b \neq 0$). Its decimal expansion either terminates (e.g. $0.75$) or repeats indefinitely (e.g. $0.\overline{3}$).
- Irrational Number ($\mathbb{R} \setminus \mathbb{Q}$) — A real number that cannot be written as a simple fraction. Its decimal representation is infinite and non-repeating ($\sqrt{2} \approx 1.41421\dots, \pi \approx 3.14159\dots, e$).
Square Roots ($\sqrt{x}$)
$$\sqrt{x} = y \iff y^2 = x \quad (y \ge 0)$$
Perfect squares: $\sqrt{1}=1, \sqrt{4}=2, \sqrt{9}=3, \dots, \sqrt{144}=12$.
Cube Roots ($\sqrt[3]{x}$)
$$\sqrt[3]{x} = y \iff y^3 = x$$
Can take cube roots of negative numbers: $\sqrt[3]{-8} = -2$ because $(-2)^3 = -8$.
§ 8.2
Axiomatic Exponent Laws & Scientific Notation
Product Rule
$$x^a \cdot x^b = x^{a+b}$$
$2^3 \cdot 2^4 = 2^7 = 128$. Add powers of common bases.
Quotient Rule
$$\frac{x^a}{x^b} = x^{a-b} \quad (x \neq 0)$$
$\frac{5^8}{5^5} = 5^{8-5} = 5^3 = 125$. Subtract powers.
Power of a Power
$$(x^a)^b = x^{a \cdot b}$$
$(3^2)^4 = 3^8 = 6,561$. Multiply outer exponent.
Zero & Negative Exponents
$$x^0 = 1 \quad \text{and} \quad x^{-n} = \frac{1}{x^n}$$
Any non-zero base to power 0 equals 1. Negative exponent flips to denominator.
DEFINITION 8.2.1
Topic: Scientific Notation Standard
$$N = a \times 10^b \quad \text{where } 1 \le |a| < 10 \text{ and } b \in \mathbb{Z}$$
$$\text{Multiplication: } (3 \times 10^4)(2 \times 10^5) = (3 \times 2) \times 10^{4+5} = \mathbf{6 \times 10^9}$$
§ 8.3
Linear Functions: Slope ($m$) & Slope-Intercept Form
The Slope Formula ($m$)
$$m = \frac{\Delta y}{\Delta x} = \frac{y_2 - y_1}{x_2 - x_1} = \frac{\text{Rise}}{\text{Run}}$$
Rate of change between two coordinates $(x_1, y_1)$ and $(x_2, y_2)$.
Slope-Intercept Form
$$y = mx + b$$
$m = \text{slope}, \quad b = y\text{-intercept at coordinate } (0, b)$.
Point-Slope Equation
$$y - y_1 = m(x - x_1)$$
Construct line directly from slope $m$ and any passing point $(x_1, y_1)$.
§ 8.4
Systems of Linear Equations (Number of Solutions)
| System Type | Geometric Configuration | Slopes & Intercepts | Solution Set |
|---|---|---|---|
| Intersecting Lines | Cross at exactly 1 intersection point | Different slopes ($m_1 \neq m_2$) | Exactly 1 Unique $(x, y)$ |
| Parallel Lines | Lines never intersect ($\parallel$) | Same slope ($m_1 = m_2$), different $b$ | No Solution ($\emptyset$) |
| Coincident Lines | Identical line plotted on top of itself | Same slope and same $y$-intercept | Infinitely Many Solutions |
EXEMPLUM 8.4
Solving a Linear System by Substitution
Problem: Solve the linear system: $\begin{cases} y = 2x + 1 \\ 3x + y = 16 \end{cases}$
Step 1 (Substitute for $y$): Replace $y$ with $(2x + 1)$ in the second equation: $3x + (2x + 1) = 16$.
Step 2 (Combine & Solve for $x$): $5x + 1 = 16 \implies 5x = 15 \implies x = 3$.
Step 3 (Solve for $y$): $y = 2(3) + 1 = 6 + 1 = 7$.
Step 4 (Verify in Equation 2): $3(3) + 7 = 9 + 7 = 16$ ✓.
Result: Unique Solution: $(x, y) = \mathbf{(3, 7)}$.
■ Q.E.D.
§ 8.5
The Pythagorean Theorem & Euclidean Distance Formula
THEOREM 8.5.1
Topic: The Pythagorean Metric
In any right triangle with perpendicular legs $a, b$ and hypotenuse $c$:
$$a^2 + b^2 = c^2 \implies c = \sqrt{a^2 + b^2}$$
$$\text{Coordinate Distance Formula: } d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}$$
§ 8.6
3D Curved Solid Volumes: Cylinders, Cones & Spheres
Volume of a Cylinder
$$V = \pi r^2 h$$
Base circle area extruded through height $h$.
Volume of a Cone
$$V = \frac{1}{3}\pi r^2 h$$
Exactly $\frac{1}{3}$ the volume of a cylinder with identical $r$ and $h$.
Volume of a Sphere
$$V = \frac{4}{3}\pi r^3$$
Enclosed volume of a complete sphere of radius $r$.
§ 8.7
Geometric Transformations: Congruence & Similarity
| Transformation | Coordinate Mapping Rule | Preserves Size / Shape? | Resulting Relationship |
|---|---|---|---|
| Translation | $(x, y) \to (x + h, y + k)$ | Preserves both | Congruent ($\cong$) |
| Reflection ($x$-axis) | $(x, y) \to (x, -y)$ | Preserves both | Congruent ($\cong$) |
| Reflection ($y$-axis) | $(x, y) \to (-x, y)$ | Preserves both | Congruent ($\cong$) |
| Rotation ($90^\circ$ CCW) | $(x, y) \to (-y, x)$ | Preserves both | Congruent ($\cong$) |
| Rotation ($180^\circ$) | $(x, y) \to (-x, -y)$ | Preserves both | Congruent ($\cong$) |
| Dilation (Scale $k$) | $(x, y) \to (kx, ky)$ | Preserves shape; changes size | Similar ($\sim$) |
§ 8.8
Bivariate Data: Scatter Plots & Lines of Best Fit
Interpreting Scatter Plot Associations
- Positive Association: As $x$ increases, $y$ tends to increase (slopes up to the right).
- Negative Association: As $x$ increases, $y$ tends to decrease (slopes down to the right).
- No Association: Data points are scattered randomly with zero trend.
- Line of Best Fit: A straight line modeled through the centroid of the cloud to make predictions.
§ 8.9
Parallel Lines Cut by a Transversal
| Angle Pair Classification | Location Relative to Transversal | Geometric Equality |
|---|---|---|
| Corresponding Angles | Same relative corner position at each intersection | Congruent ($=$) |
| Alternate Interior | Opposite sides of transversal, between parallel lines | Congruent ($=$) |
| Alternate Exterior | Opposite sides of transversal, outside parallel lines | Congruent ($=$) |
| Consecutive Interior | Same side of transversal, between parallel lines | Supplementary ($\Sigma = 180^\circ$) |
§ 8.10
Scholia & Computational Mental Math Speed Hacks
HACK I
Pythagorean Triples Memory Bank
$$\mathbf{3-4-5} \quad \vert \quad \mathbf{5-12-13} \quad \vert \quad \mathbf{8-15-17} \quad \vert \quad \mathbf{7-24-25}$$
Multiples work too: $(6-8-10, 9-12-15)$. If legs are $6$ and $8$, hypotenuse is $10$ in $0.5$ seconds!
HACK II
"Rise over Run" Slope Mantra
$$m = \frac{\Delta y}{\Delta x} = \frac{\text{Vertical Change}}{\text{Horizontal Change}}$$
Remember: $Y$ flies high into the sky (numerator); $X$ runs on the train tracks (denominator).
HACK III
Negative Exponent Reciprocal Flip
$$\left(\frac{a}{b}\right)^{-n} = \left(\frac{b}{a}\right)^n$$
Flip the entire fraction to make the power positive! $\left(\frac{2}{3}\right)^{-3} = \left(\frac{3}{2}\right)^3 = \frac{27}{8}$.
HACK IV
Cylinder vs. Cone 3:1 Volume Ratio
$$V_{\text{cylinder}} = 3 \times V_{\text{cone}}$$
It takes exactly 3 full cones of liquid to fill 1 cylinder of the same radius and height!
§ 8.11
The Grand Synoptic Tables & Complete Grade 8 Student Cheat Sheet
Grade 8 Master Reference Concordance
Authorized curriculum reference concordance • Grade 8 Standards Complete
Exponent Laws
- $x^a \cdot x^b = x^{a+b}$
- $x^a / x^b = x^{a-b}$
- $(x^a)^b = x^{ab}$
- $x^0 = 1 \quad (x \neq 0)$
- $x^{-n} = 1/x^n$
- Scientific: $a \times 10^b$ ($1 \le a < 10$)
Linear Functions
- Slope: $m = \frac{y_2-y_1}{x_2-x_1}$
- Slope-Intercept: $y = mx + b$
- Point-Slope: $y - y_1 = m(x - x_1)$
- Parallel: $m_1 = m_2$
- Perpendicular: $m_1 \cdot m_2 = -1$
- Horizontal: $m = 0$ • Vertical: Undefined
Linear Systems
- Intersecting ($m_1 \neq m_2$): 1 Solution
- Parallel ($m_1 = m_2, b_1 \neq b_2$): No Solution
- Coincident ($m_1=m_2, b_1=b_2$): $\infty$ Solutions
- Methods: Graphing, Substitution, Elimination
Pythagoras & Distance
- $a^2 + b^2 = c^2 \implies c = \sqrt{a^2 + b^2}$
- $d = \sqrt{(\Delta x)^2 + (\Delta y)^2}$
- Triples: $3-4-5, 5-12-13$
- Triples: $8-15-17, 7-24-25$
- Converse: If $a^2+b^2=c^2 \implies 90^\circ$
3D Curved Volumes
- Cylinder: $V = \pi r^2 h$
- Cone: $V = \frac{1}{3}\pi r^2 h$
- Sphere: $V = \frac{4}{3}\pi r^3$
- Cylinder to Cone Ratio: $3 : 1$
- $\pi \approx 3.14159 \approx \frac{22}{7}$
Transformations
- Translation: $(x+h, y+k)$ (Congruent)
- Reflect $x$-axis: $(x, -y)$
- Reflect $y$-axis: $(-x, y)$
- Rotate $90^\circ$ CCW: $(-y, x)$
- Rotate $180^\circ$: $(-x, -y)$
- Dilation: $(kx, ky)$ (Similar)