# Quadratic Functions and Parabolas

• Parabolas

• Quadratic equations and functions

• Graphs of quadratic functions

• Applications

** Quadratic Functions and Expressions
**A quadratic function has two forms:

• f(x) = ax^2 +bx+c (standard form)

• f(x) = a(x − h)^2 +k (vertex-axis form)

The graph of a quadratic function is a parabola. It is easy to

graph a quadratic function if it is expressed in the vertex-axis

form.

**Graphing a quadratic function**

The vertex is the point at (2, 4)

The axis of symmetry is the vertical line x = 2

**Graphing a quadratic function**

y = 2(x+3)^2 − 5

The vertex is the point at

The axis of symmetry is the vertical line

**Graphing a quadratic function**

The general case

y = a(x − h)^2 +k

The general case

The vertex is the point at (h, k)

The axis of symmetry is the vertical line

If a > 0, the parabola opens upward.

If a < 0, the parabola opens downward.

Questions:

• Does the graph of f(x) = 5(x − 1)^2 + 8 open upward or

downward?

• Does the graph of open upward or

downward?

• What is the equation for the axis of symmetry for the graph

of f(x) = 5(x − 1)^2 +8?

• What are the coordinates of the vertex of the graph of ?

**Completing the square**

How to change the standard form for the function into the vertexaxis

form.

**Example:**f(x) = x^2 − 6x+10

Change into vertex -axis form.

**Change to vertex-axis form by completing the square:**

Problem:

Problem:

f(x) = x^2 +4x − 5

**Problem:**Change to vertex-axis form by completing the square:

f(x) = −x^2 − 10x+1

**Change to vertex-axis form by completing the square:**

Problem:

Problem:

f(x) = 3x^2 +6x+1

**Problem:**Change to vertex-axis form by completing the square:

f(x) = 5x^2 − 30x+11

The general quadratic function:

f(x) = ax^2 +bx+c

The quadratic formula tells you the solutions to f (x) = 0,

which is the same as locating the x- intercepts on the graph :

**
Example:** Solve

2x^2 − 5x −3 = 0,

for x.

a = 2, b= −5, c = −3

So x = 3 and x = 1/2 are the solutions .

**Example:**Solve

x^2 − 5x −5 = 0,

for x.

a = , b= , c =

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