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Chapter 2: Quadratic Functions
- Lesson 1: Transformations of Quadratic Functions
- Quadratic Functions
- A quadratic function is a function that can be written in the form f(x) = a(x - h)2 + k, where a ≠ 0
- The U-shaped graph of a quadratic function is called a parabola
- Translations
- Horizontal Translation
- f(x) = x2
- f(x - h) = (x - h)2
- Shifts left when h < 0
- Shifts right when h > 0
- Vertical Translation
- f(x) = x2
- f(x) + k = x2 + k
- Shifts down when k < 0
- Shifts up when k > 0
- Reflections
- Reflection in the x-axis
- f(x) = x2
- -f(x) = -(x2) = -x2
- Reflection in the y-axis
- f(x) = x2
- f(-x) = (-x)2 = x2
- Stretches and Shrinks
- Horizontal Stretches and Shrinks
- f(x) = x2
- f(ax) = (ax)2
- Horizontal stretch (away from y-axis) when 0 < a < 1
- Horizontal shrink (towards y-axis) when a > 1
- Vertical Stretches and Shrinks
- f(x) = x2
- a * f(x) = ax2
- Vertical stretch (away from x-axis) when a > 1
- Vertical shrink (towards x-axis) when 0 < a < 1
- Vertex Form
- The lowest part on a parabola that opens up or the highest point on a parabola that opens down is called the vertex
- The vertex form of a quadratic function is f(x) = a(x - h)2 + k, where a ≠ 0 and the vertex is (h, k)
- Lesson 2: Characteristics of Quadratic Functions
- Quadratic Function Forms
- Axis of Symmetry - a line that divides a parabola into mirror images and passes through the vertex; the axis of symmetry is the vertical line x = h or x = -b/2a
- Standard Form - a quadratic function written in the form f(x) = ax2 + bx + c
- Intercept Form - a quadratic function in the form f(x) = a(x - p)(x - q)
- Properties of the Standard Form
- The parabola opens up when a > 0 and opens down when a < 0
- The graph is narrower than the graph of f(x) = x2 when |a| > 1 and wider when |a| < 1
- The axis of symmetry is x = -b/2a and the vertex is (-b/2a, f(-b/2a))
- The y-intercept is c; the point (0, c) is on the parabola
- Minimum and Maximum Values
- For the quadratic function f(x) = ax2 + bx + c, the y-coordinate of the vertex is the minimum value of the function when a > 0 and the maximum value when a < 0
- a > 0
- Minimum value: f(-b/2a)
- Domain: all real numbers
- Range: y ≥ f(-b/2a)
- Decreasing to the left of x = -b/2a
- Increasing to the right of x = -b/2a
- a < 0
- Maximum value: f(-b/2a)
- Domain: all real numbers
- Range: y ≤ f(-b/2a)
- Increasing to the left of x = -b/2a
- Decreasing to the right of x = -b/2a
- Properties of the Intercept Form
- Because f(p) = 0 and f(q) = 0, p and q are the x-intercepts of the graph of the function
- The axis of symmetry is halfway between (p, 0) and (q, 0); x = (p + q)/2
- The parabola opens up when a > 0 and opens down when a < 0
- Lesson 3: Modeling with Quadratic Equations
- Writing Quadratic Equations
- Given a point and the vertex (h, k), use vertex form: y = a(x - h)2 + k
- Given a point and x-intercepts p and q, use intercept form y = a(x - p)(x - q)
- Given three points, write and solve a system of three equations in three variables
Note
5.0(1)
Explore Top NotesNoteStudied by 8 peopleNoteStudied by 13 peopleNoteStudied by 152 peopleNoteStudied by 13 peopleNoteStudied by 39 peopleNoteStudied by 16 people
House + Home
NoteStudied by 8 peopleLecture 4
NoteStudied by 13 peopleIntroduction to the Legal System
NoteStudied by 152 peopleSocial and Class Relations (Prehistoric Era to 600 CE)
NoteStudied by 13 peopleVocabulario para describir una persona
NoteStudied by 39 peopleAkkad, Mesopotamia's First Empire
NoteStudied by 16 people