Unimaginable tricks on Expressing a Quadratic Function to Vertex Form, a[(x+b)²+c] скачать в хорошем качестве

Unimaginable tricks on Expressing a Quadratic Function to Vertex Form, a[(x+b)²+c]

After watching this video, you would be able rewriting the Quadratic Function, f(x) = 5x²+20x-16 to Vertex Form, f(x)=a[(x+b)²+c]. That is; by method of completing the squares. Quadratic Function A quadratic function is a polynomial function of degree 2, represented in the form: f(x) = ax^2 + bx + c where: \(a\), \(b\), and \(c\) are constants, \(a
eq 0\). Key Features: 1. *Graph*: The graph of a quadratic function is a parabola. 2. *Vertex*: The vertex is the highest or lowest point on the parabola, given by: h = -\frac{b}{2a} k = f(h) The vertex form is \( f(x) = a(x - h)^2 + k \). 3. *Axis of Symmetry*: The vertical line \( x = h \) that passes through the vertex. 4. *Roots/Zeros*: The values of \(x\) where \(f(x) = 0\), found using the quadratic formula: x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} Example: Consider the quadratic function \( f(x) = x^2 - 4x + 4 \). 1. *Vertex*: h = -\frac{-4}{2(1)} = 2 k = f(2) = (2)^2 - 4(2) + 4 = 0 Vertex: \((2, 0)\) 2. *Axis of Symmetry*: \( x = 2 \) 3. *Roots*: x = \frac{4 \pm \sqrt{(-4)^2 - 4(1)(4)}}{2(1)} = 2 Root: \( x = 2 \) Applications: Physics: Projectile motion. Engineering: Designing parabolic structures. Economics: Modeling profit and cost functions. Converting a Quadratic Function to Vertex Form The vertex form of a quadratic function is given by: y = a(x - h)^2 + k where \((h, k)\) is the vertex of the parabola. Steps to Convert Standard Form to Vertex Form: 1. *Start with the standard form*: \(y = ax^2 + bx + c\) 2. *Factor out \(a\) from the first two terms*: \(y = a(x^2 + \frac{b}{a}x) + c\) 3. *Complete the square*: Add and subtract \((\frac{b}{2a})^2\) inside the parenthesis: y = a(x^2 + \frac{b}{a}x + (\frac{b}{2a})^2 - (\frac{b}{2a})^2) + c y = a((x + \frac{b}{2a})^2 - (\frac{b}{2a})^2) + c 4. *Simplify*: y = a(x + \frac{b}{2a})^2 - a(\frac{b}{2a})^2 + c y = a(x + \frac{b}{2a})^2 - \frac{b^2}{4a} + c 5. *Write in vertex form*: y = a(x - h)^2 + k where h = -\frac{b}{2a}\) and \(k = c - \frac{b^2}{4a}\). Example: Convert y = 2x^2 + 8x + 5 to vertex form. 1. *Factor out 2*: y = 2(x^2 + 4x) + 5 2. *Complete the square*: y = 2(x^2 + 4x + 4 - 4) + 5 y = 2((x + 2)^2 - 4) + 5 3. *Simplify*: y = 2(x + 2)^2 - 8 + 5 y = 2(x + 2)^2 - 3 The vertex form is y = 2(x + 2)^2 - 3, with vertex (-2, -3) Would you like to watch other similar videos on expressing quadratic functions to vertex form?