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
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