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Quadratic Functions in Standard Form

Sep 17, 2022
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Key Concepts

  • Define the standard form of a quadratic function.
  • Find the Y-intercept of a quadratic function.
  • Find the axis of symmetry of a quadratic function.
  • Find the vertex of a quadratic function.

Quadratic function 

A function f defined by f(x) = ax2+bx+c, where  a, b and c are real numbers and  a≠0, is called a quadratic function

  • The graph of a quadratic function is a curve called a parabola
Quadratic function graph 1
  • The quadratic parent function is f(x) = x2
  • For 0<|a|<1, the shape of the parabola is wider than the parent function. 
  • For |a|>1, the shape of the parabola is narrower than the parent function. 
Quadratic function graph 2
  • f(x) = ax2 is the reflection of f(x) = −ax2 over the x-axis. 
Quadratic function graph 3

Vertex form of the quadratic function 

  • The function f(x) = a(x−h)2+k, where a≠0 is called the vertex form of the quadratic function
Vertex form of the quadratic function
  • The vertex of the graph g is (h, k). 
  • The graph of f(x) = ax−h2+k is a translation of the function f(x) = ax2 that is translated h units horizontally and k units vertically. 
  • The value of a does not affect the location of the vertex. 

Graph of g(x) = x2+k

The value of k in g(x) = x2+k translates the graph of parent function f, vertically k units. The value of k does not affect the axis of symmetry. 

Graph of g(x) = x2+k

Graph of g(x) = (x−h)

  • The value of h in g(x) = (x−h)2 translates the graph of parent function f, horizontally h units. 
  • The vertex of the graph g is (0, h). 
Graph of g(x) = (x−h)2
  • The value of h translates the axis of symmetry. 

A standard form of a quadratic function 

  • The standard form of a quadratic function is ax2+bx+c=0, where a≠0 
  • The axis of symmetry of a standard form of quadratic function f(x) = ax2+bx+c is the line x=−b/2a. 
  • The y-intercept of f(x) is c. 
  • The x-coordinate of the graph of f(x) = ax2+bx+c is –b/2a. 

The vertex of f(x) = ax2+bx+c is (-b/2a, f(-b/2a))

Exercise

  • Find the intercept, x intercept, axis of symmetry, and vertex of the graph of f(x) = 3x2 + 6x + 3
  • Graph g(x) = x2 + 2x – 7

What have we learned

  • The standard form of a quadratic function is ax2+bx+c=0, where a≠0 

Concept Map 

Concept Map
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