Quadratic Functions in Standard Form

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. 

• 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. 

• f(x) = ax² is the reflection of f(x) = −ax² over the x-axis. 

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. 

• 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) = (x−h)² 

• 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). 

• 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