Welcome to MPM2D0 

Exploring the Vertex Form of a Quadratic Relationship.

In the following equation: y= ax^2 

We are going to determine the effect of a in the equation.

 

Examples: y= 3x^2, y= 0.5x^2, and y= -0.25x^2

 

When graphing the following equations. Many comparisons are established:

 

Firstly, when graphing the equations y=3x^2 and y= 0.5x^2. The effect of 3 as the a value in y=ax^2, the parabola is very skinny and close to the y- intercept in other terms it is vertically stretched, while the effect of 0.5 as the a value, the parabola is bigger and away from y axis meaning it is being vertically compressed. 

 

However, when graphing y= -0.25x^2, the parabola is vertically compressed, but since there is a negative in from the a value the parabola is reflected on the x axis and now opens down. 

 

Note: The effect of a in the y=ax^2 is that the value of a determine if the parabola is vertically stretched or compressed and the sign in front of the a value determines whether the parabola opens up or down.

 

Now we explore effect of k in y= x^2 + k

 

We learn that when you if the k value is positive, that is the amount of units the vertex moves up and if the k value is a negative, the vertex moves down certain units. 

 

Lastly we are going to explore the effect of h in y= (x-h)^2

 

The effect of h value on the vertex and parabola itself is that it determines if the vertex moves left or right. If the h value is negative the vertex moves to the left and if the h value is positive the vertex moves to the right. 

 

Recall: 

example: y= 2 (x-3)^2 +5 

The number 2 means that the parabola is stretched by the factor of 2.

The number -3 means that the vertex moves 3 units to the left.

The number 5 means that the vertex moves 5 units up. 

 

• The a value determines if the vertex is being compressed or stretched. 

• The k value determines whether the vertex moves up or down.

• The h value determines whether the vertex moves right or left. 

 

 

 

Transformations of a Parabola.

 

 A parabola takes many transformations. However, the vertex form of a parabola is y = a (x-h)^2 +k 

 

The basic parabola has the formula of y= x^2

 

As discussed before the value of "a" determines the orientation and the shape of the parabola. 

 

To determine the orientation of the parabola: 

 

• If the a value is greater than 0, the parabola must open up. ( like a smile)

• If the a value is less than 0 , the parabola must open down. ( like a frown) 

 

To determine the shape of the parabola: 

 

• If -1 < a < 1, the parabola is vertically compressed.

• if a > 1 or a < -1, the parabola is vertically stretched.

 

 Example;  2x^2 

-opens up 

- vertically stretched by a factor of 2.

 

Example 2) -9x^2

 

- opens down.

- stretched by a factor of 9.

 

Example 3) - 0.25x^2

 

-opens down.

- vertically compressed by a factor of 0.25.

 

The value of "k" determine the vertical position of the parabola.

 

• If k > 0, the vertex moves up by the k units.

• If k < 0, the vertex moves down by the k units. 

 

Example; y= x^2+3

- 3 units up.

 

Example 2) y= x^2 -6

-6 units down.

 

The value of h determine the horizontal position of the parabola. 

 

• If h > 0, the vertex moves to the right h units.

• If h < 0, the vertex moves to the left h units.