Mishita Dharia - Quadratics

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

 

A parabola can go through 3 different transformations that can affect its orientation and shape, vertical position, and horizontal position.

 

The a-value of a parabola can determine the orientation and shape of the parabola.

 

If the a-value is greater than 0 (positive) then the parabola will open up.If the a-value is less than 0 (negative) then the parabola will open down (reflection about x-axis).

 

The parabola would be vertically compressed if -1<a<0 or 0<a<1.

E.g. the parabola would be vertically compressed if the a value were 0.5, -0.3, ¼.

 

The parabola would be vertically stretched if a > 1 or a < -1.

E.g. the parabola would be vertically stretched if the a value were 3, -1.5, 4.5, -5.

 

For example if the equation were y = 6x^2 you would write:

• Opens up

• Vertically stretched by a factor of 6.

 

For example if the equation were y = -0.5x^2 you would write:

• Opens down (reflection about x-axis).

• Vertically compressed by a factor of 0.5.

 

The k-value of a parabola can determine the vertical position of the parabola.

 

If k > 0 the vertex would move up k units.

 

For example if the equation were y = x^2 + 5 you would write:

• Translated 5 units up.

• Vertex (0, 5)

 

If k < 0 the vertex would move down k units.

 

For example if the equation were y = x^2 - 7 you would write:

• Translated 7 units down.

• Vertex (0, -7)

 

The h-value of a parabola can determine the horizontal position of the parabola.

 

When identifying the h-value in an equation, make sure that you switch the signs of the number, since the number would have been affected by the negative sign in front of the h in the original equation.

 

Ex. In the equation y = (x – 6)^2 although the h-value looks like it is negative 6, it is actually positive 6 since when put into the equation in place of the h it was converted to -6 because of the negative sign in the original equation. Similarly in the equation y = (x + 6)^2 the h-value would be negative 6 due to the negative sign already in the equation. Negative x negative = positive, and positive x negative = negative.

 

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

 

For example if the equation were y = (x-6)^2 and h=6, you would write:

• Translated 6 units to the right.

 

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

 

For example if the equation were y = (x + 7)^2 and h=6, you would write:

• Translated 7 units to the left.

 

 

Putting it all Together:

If you were to have the equation of y = 18(x + 7)^2 +5, you would first identify the a, h, and k values.

a = 18 h = -7 k = 5

 

Then you would write how these values would transform the parabola.

• Opens up

• Vertically stretched by a factor of 18

• Translated 7 units to the left

• Translated 5 units up

• Vertex (-7, 5)

 

Now if you are given the transformations and you have to write the equation, you would just go backwards…

 

Example: Write the Equation Given the Transformations

• Opens down (reflection about x-axis)

• Vertically compressed by a factor of 0.9

• Translated 1 unit to the right

• Translated 8 units down

 

1. Identify the a-value. Remember that its orientation affects the sign in front of the a-value.

a = -0.9

2. Identify the h- value, the horizontal translation. Remember to switch the signs of the h-value when putting it into the equation.

h = 1

3. Identify the k-value, the vertical translation.

k = 8

4. Put it all together.

y =0.9(x-1)2 -8