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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) = ax^{2}+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) = x
^{2}. - 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) = ax
^{2}is the reflection of f(x) = −ax^{2}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−h
^{2}+k is a translation of the function f(x) = ax^{2}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) = x^{2}+k

The value of k in g(x) = x^{2}+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 ax
^{2}+bx+c=0, where a≠0 - The axis of symmetry of a standard form of quadratic function f(x) = ax
^{2}+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) = ax
^{2}+bx+c is –b/2a.

The vertex of f(x) = ax^{2}+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) = 3x
^{2}+ 6x + 3 - Graph g(x) = x
^{2}+ 2x – 7

### What have we learned

- The standard form of a quadratic function is ax
^{2}+bx+c=0, where a≠0

### Concept Map

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