Week_3_Quadratic_Functions

Week 3: Quadratic Functions & Equations

1. Quadratic Functions

A quadratic function is a polynomial of degree 2: $f(x)=a{x}^2+bx+c$ where $a,b,c\in \mathbb{R}$ and $a\ne 0$.

1.1 The Graph: Parabola

• Concavity:
  • If $a>0$: Opens upward (Valley). Has a minimum.
  • If $a<0$: Opens downward (Hill). Has a maximum.
• Vertex $(h,k)$: The turning point of the parabola.
  • x-coordinate: $h=-\frac{b}{2a}$.
  • y-coordinate: $k=f(h)=c-\frac{b^2}{4a}=-\frac{D}{4a}$ (where $D$ is the discriminant).
• Axis of Symmetry: The vertical line $x=-\frac{b}{2a}$.

1.2 Vertex Form

Any quadratic can be rewritten as: $f(x)=a(x-h{)}^2+k$

• This form instantly tells you the vertex $(h,k)$.
• Conversion: Complete the square.
  • $f(x)=2x^2+8x+5\to 2(x^2+4x)+5\to 2(x^2+4x+4-4)+5\to 2(x+2{)}^2-8+5\to 2(x+2{)}^2-3$. Vertex is $(-2,-3)$.

---

2. Quadratic Equations

Solving $a{x}^2+bx+c=0$.

2.1 The Quadratic Formula

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

2.2 Methods for Solving

1. Factoring: Splitting the middle term. e.g., $x^2+5x+6=0\to (x+2)(x+3)=0$.
2. Completing the Square: Transforming to $(x+p{)}^2=q$.
3. Quadratic Formula: Universal method.

2.3 Nature of Roots (Discriminant)

Let $D=b^2-4ac$.

• $D>0$: Two distinct real roots. (Graph cuts x-axis twice).
• $D=0$: Exactly one real root (repeated root). (Graph touches x-axis at vertex).
• $D<0$: No real roots (Complex roots). (Graph does not touch x-axis).

2.4 Sum and Product of Roots

If $\alpha ,\beta$ are roots:

• Sum: $\alpha +\beta =-\frac{b}{a}$
• Product: $\alpha \beta =\frac{c}{a}$

---

3. Optimization Problems

Since the vertex represents the max or min, quadratics are used to optimize values.

• Max/Min Value: Always occurs at $x=-b/2a$.
• Value: The actual max/min value is $f(-b/2a)$.

Tacit Knowledge:

In word problems (e.g., “maximize area of a rectangle with fixed perimeter”), assume variables $x$ and $y$. Relate them ($2x+2y=P$). Express area $A=xy$ in terms of one variable ($A=x(P/2-x)$) to get a quadratic. Find the vertex.

---

4. Goated Examples

Example 1: Roots and Coefficients

Question: If roots of $2x^2-kx+8=0$ differ by 2, find $k$. Solution:

1. Roots $\alpha ,\beta$. Given $|\alpha -\beta |=2$.
2. Sum: $\alpha +\beta =k/2$.
3. Product: $\alpha \beta =8/2=4$.
4. Identity: $(\alpha -\beta {)}^2=(\alpha +\beta {)}^2-4\alpha \beta$.
5. $2^2=(k/2{)}^2-4(4)⟹4=\frac{k^2}{4}-16⟹20=\frac{k^2}{4}⟹k^2=80⟹k=\pm 4\sqrt{5}$.

Example 2: Range of Functions

Question: Find range of $f(x)=x^2-4x+7$ for $x\in \mathbb{R}$. Solution:

1. $a=1>0$, so it opens up. Min value is at vertex.
2. $x_{vertex}=-(-4)/2(1)=2$.
3. $y_{min}=f(2)=2^2-4(2)+7=4-8+7=3$.
4. Range: $[3,\infty )$. Trap: If domain is restricted (e.g., $x\in [-1,1]$), check endpoints!
   • $f(-1)=1+4+7=12$.
   • $f(1)=1-4+7=4$.
   • Vertex at $x=2$ is outside $[-1,1]$.
   • Min is at $f(1)=4$, Max is at $f(-1)=12$. Range: $[4,12]$.

Example 3: Intersection of Line and Parabola

Question: For what $c$ does $y=x+c$ touch $y=x^2$? Solution:

1. Equate: $x^2=x+c⟹x^2-x-c=0$.
2. “Touches” means tangent $⟹$ One solution $⟹D=0$.
3. $D=(-1{)}^2-4(1)(-c)=1+4c=0$.
4. $c=-1/4$.