Week-03

Maths 1: Week 03 - The Master Encyclopedia of Quadratic Dynamics

1. The Genesis: The Geometry of Projectiles 📜

1.1 The Curve of Nature

Long before mathematicians had a formula, they had the Curve. Whether it’s a pebble skipping across water or a planet orbiting a sun, nature moves in arcs. In the 17th century, Galileo Galilei discovered that falling objects follow a parabolic path.

This realization turned the “Parabola”—previously just a slice of a cone (Conic Section)—into the most important tool for the nascent field of Physics. The Quadratic Equation ($a{x}^2+bx+c=0$) is the algebraic soul of this physical reality. It allows us to calculate the exact peak of a flight or the exact moment of impact.

1.2 The Philosophical Intuition

A Quadratic Function is the simplest form of Non-Linear Change. While a line is constant, a quadratic is “accelerating.” It introduces us to the concept of the “Turning Point”—where a trend reverses direction.

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2. Axiomatic Foundations: The General Form 🏛️

2.1 The Standard Definition

A Quadratic Function is a function $f:\mathbb{R}\to \mathbb{R}$ of the form: $f(x)=a{x}^2+bx+c$ Where $a,b,c$ are real numbers and $a\ne 0$.

The Power of '$a$'

• If $a>0$: The function is Convex (happy face). It opens upwards and has a Global Minimum.
• If $a<0$: The function is Concave (sad face). It opens downwards and has a Global Maximum.

2.2 The Three Forms of a Quadratic

To master quadratics, you must be able to translate between these three “dialects”:

1. General Form: $f(x)=a{x}^2+bx+c$. (Best for intercept finding).
2. Vertex Form: $f(x)=a(x-h{)}^2+k$. (Best for finding the peak/valley).
3. Factored Form: $f(x)=a(x-r_1)(x-r_2)$. (Best for finding where it hits the ground).

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3. The Topology of the Parabola: The Vertex 🛠️

3.1 Deriving the Axis of Symmetry

The parabola is perfectly symmetrical. The vertical line that cuts it in half is called the Axis of Symmetry.

• Derivation by Completing the Square: Starting with $a{x}^2+bx+c$, we factor out $a$: $a[x^2+\frac{b}{a}x]+c$ To make $x^2+\frac{b}{a}x$ a perfect square, we add $(b/2a{)}^2$. This leads to the Vertex $x$-coordinate: $h=-\frac{b}{2a}$.

3.2 The Vertical Shift ($k$)

Once you have $h$, you find $k$ (the actual max/min height) by plugging $h$ into the function: $k=f\left(-\frac{b}{2a}\right)$

• The Vertex: $V(h,k)=\left(-\frac{b}{2a},\frac{4ac-b^2}{4a}\right)$.

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4. Roots & The Discriminant: Predictable Reality 🖋️

4.1 The Quadratic Formula

The roots ($x$-intercepts) of $a{x}^2+bx+c=0$ are: $x=\frac{-b\pm \sqrt{b^2-4ac}}{2a}$

4.2 The Role of the Discriminant ($D$)

The term under the radical, $D=b^2-4ac$, is the “Fortune Teller” of the parabola:

1. $D>0$: The parabola crosses the $X$-axis twice. (2 Real Roots).
2. $D=0$: The parabola just “kisses” the $X$-axis at the vertex. (1 Real Root).
3. $D<0$: The parabola is “floating” in space. It never touches the $X$-axis. (No Real Roots).

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5. Optimization: Peak Performance 📈

5.1 The Logic of the Extreme

In real world models (Economics, Physics, Engineering), the “Best” outcome usually happens at the vertex.

• Recipe for Optimization:
  1. Identify the quadratic relationship.
  2. Verify if you need a Max ($a<0$) or Min ($a>0$).
  3. Calculate $x=-b/2a$.
  4. The peak result is $f(x)$.

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6. The Encyclopedia of Worked Examples (10 Case Studies) 📚

Case 1: Converting to Vertex Form

Problem: Convert $f(x)=2x^2-8x+10$ to vertex form.

• Thinking Process: Factor out $a$ from the $x$ terms. Find the “missing piece” to complete the square.
• Calculation:
  • $f(x)=2(x^2-4x)+10$
  • Inside the bracket, to complete $x^2-4x$, we need $(-4/2{)}^2=4$.
  • Add and subtract 4 inside: $2(x^2-4x+4-4)+10$
  • Bring out the $-4$ (don’t forget to multiply by 2): $2(x-2{)}^2-8+10$.
• Result: $f(x)=2(x-2{)}^2+2$. Vertex is $(2,2)$.

Case 2: The Projectile Problem (Max Height)

Problem: A missile follows $H(t)=-16t^2+64t+10$. What is the max height?

• Step 1: $a=-16,b=64$.
• Step 2: $t=-64/(2\times -16)=2$ seconds.
• Step 3: $H(2)=-16(4)+64(2)+10=-64+128+10=74$.
• Result: Max height is 74 meters.

Case 3: Finding $c$ for a Tangent Line

Problem: For what value of $c$ does the line $y=4x$ just touch $y=x^2+c$?

• Thinking Process: If they touch at one point, the equation $x^2+c=4x⟹x^2-4x+c=0$ must have $D=0$.
• Calculation: $b^2-4ac=(-4{)}^2-4(1)(c)=0⟹16=4c⟹c=4$.

Case 4: Range of a Quadratic

Problem: Find the range of $f(x)=x^2-4x+5$.

• Step 1: Since $a=1>0$, the range is $[k,\infty )$.
• Step 2: Vertex $x=-(-4)/2=2$.
• Step 3: $k=f(2)=4-8+5=1$.
• Result: Range is $[1,\infty )$.

Case 5: Root Sum and Product Rules (Vieta’s)

Problem: For $2x^2+10x+5=0$, find the sum and product of roots without solving.

• Identity: Sum $=-b/a$. Product $=c/a$.
• Result: Sum $=-10/2=-5$. Product $=5/2=2.5$.

Case 6: Intersection of Parabola and Line

Problem: Find where $y=x^2$ meets $y=x+2$.

• Calculation: $x^2=x+2⟹x^2-x-2=0⟹(x-2)(x+1)=0$.
• Result: Points $(2,4)$ and $(-1,1)$.

Case 7: Symmetry Proof

Problem: Prove $f(3)$ and $f(1)$ are equal for $f(x)=x^2-4x+7$.

• Logic: Vertex is $x=2$. $x=3$ and $x=1$ are equidistant from the vertex (distance 1). By symmetry, they must have the same height.

Case 8: The Area under a Scaffolding

Problem: A quadratic arch $y=-x^2+9$ is placed over the $X$-axis. Find its width at the base.

• Logic: Width at base is the distance between roots.
• Calculation: $-x^2+9=0⟹x^2=9⟹x=\pm 3$.
• Result: Width $=3-(-3)=6$.

Case 9: Slope of a Quadratic

Problem: What is the instantaneous slope formula for $y=a{x}^2+bx+c$?

• Derivation: using limits (introduced next week) or basic power rule: $2ax+\mathbf{b}$.

Case 10: The Reflective Property (Physics)

Problem: Why are satellite dishes parabolic?

• Geometrical Fact: All parallel rays entering a parabola reflect and pass through a single point called the Focus. This concentrates the signal.

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7. Fundamental “How-To” Recipes 🍳

Recipe: Rapid Graphing of a Parabola

1. Vertex: Find $(h,k)$. Plot it.
2. Y-intercept: Plug $x=0$. Plot it.
3. Symmetry: Plot the reflection of the Y-intercept across the $x=h$ line.
4. Roots: Solve for $D$. If $D>0$, plot the two roots.
5. Connect: Draw a smooth curve through these 5 points.

Recipe: Modeling from Word Problems

1. Identify variables: Let $x$ be the decision (Price, Time) and $y$ be the result (Profit, Height).
2. Standardize: Force the equation into $y=a{x}^2+bx+c$.
3. Identify $a$: Is it a Growth or a Peak?

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8. Encyclopedia Mastery Challenge 🏆

1. The Floating Vertex: If $k>0$ and $a>0$, can the quadratic have real roots? Explain.
2. The Vieta Proof: Take the quadratic formula roots $\frac{-b+\sqrt{D}}{2a}$ and $\frac{-b-\sqrt{D}}{2a}$ and add them. Does it equal $-b/a$? Show your steps.
3. The Shift Challenge: If you shift $y=x^2$ three units right and five units up, what is the new vertex form equation?
4. The Triple Point Check: How many points do you need to uniquely define a parabola? (Hint: Think about linear systems).

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🚀 Master Status: You have completed the Encyclopedic Expansion of Quadratic Dynamics. You can now predict the peaks and valleys of any accelerating system.