﻿ Standard and vertex form of the equation of parabola and how it relates to a parabola's graph.

# Equation of a Parabola

Standard Form and Vertex Form Equations

The equation of a parabola can be expressed in either standard or vertex form as shown in the picture below.

# Standard Form Equation

The standard form of a parabola's equation is generally expressed:

$$y = ax^2 + bx + c$$

The role of 'a'

The role of 'a'

The larger the $|a|$ is (when $|a|$ is greater than 1), the more the graphs narrows.

Case I : When $|a| > 1$

Case II : When $|a| < 1$

The larger the $|a|$ is (when $|a|$ is greater than 1), the more the graphs narrows.

The axis of symmetry

The axis of symmetry is the line $x = -\frac{b}{2a}$

Picture of Standard form equation
Axis of Symmetry from Standard Form

# Vertex Form

The vertex form of a parabola's equation is generally expressed as: $y = a(x-h)^2 +k$

• (h,k) is the vertex as you can see in the picture below
• If a is positive then the parabola opens upwards like a regular "U".
• If a is negative, then the graph opens downwards like an upside down "U".
• And, just like standard form, the larger the $|a|$, the more narrow the parabola's graph gets.

The role of 'a'

Case I : When $|a| > 1$

The larger the $|a|$ is, the more the graphs narrows.

Case II : When $|a| < 1$

The larger the $|a|$ is, the more narrow the parabola is. Or, another way to think of it, the closer that the value of $a$ gets to zero, the wider the parabola becomes.

# Vertex Form Practice Problems

##### Problem 1

The parabola's vertex is the point (1,1).

#### Identifying the vertex in vertex form

##### Problem 4.1

The vertex is the point (-3,4)

##### Problem 4.2

(3,4) is the vertex.

##### Problem 4.3
vertex is (2, –3)

#### Part II

##### Problem 5.1

The vertex is (3,4) and it opens upwards since a is positive( it is 2), it opens upwards.

##### Problem 5.2

Vertex = (-3, 4), and it opens upwards since a is positive.

##### Problem 5.3

Vertex = (9, 5) and since a is negative (it is -22), it opens downwards.