The vertex of the parabola is at (h, k).

Vertex form of a quadratic can be used to determine transformations of the quadratic parent function. Quadratic parent function: f(x) = x2

Horizontal Translations: If f(x) = (x – 2)2 then for (x – h)2 ,(x – (2))2, h = 2. The graph moves two units to the right.

Horizontal Translations: If f(x) = (x + 3)2 then for (x – h)2 ,(x – (-3))2 , h = -3 The graph moves three units to the left.

Vertical Translations: If f(x) = (x)2 + 2 then for (x – h)2 + k, (x)2 + 2, k = 2 The graph moves two units up.

Vertical Translations: If f(x) = (x)2 – 1 then for (x – h)2 + k, (x)2 – 1, k = -1 The graph moves one unit down.

Horizontal and Vertical Translations: If f(x) = (x – 3)2 + 1 then for (x – h)2 + k, (x – (3))2 + 1, h = 3 and k = 1 The graph moves three units right and 1 unit up.

Horizontal and Vertical Translations: If f(x) = (x + 1)2 – 2 then for (x – h)2 + k, (x – (-1))2 – 2, h = -1 k = -2 The graph moves one unit left and two units down.

Horizontal and Vertical Translations: The vertex of a parabola after a translation is located at the point (h, k). If f(x) = (x + 7)2 + 3 then for (x – h)2 + k, (x – (-7))2 + 3, h = -7 k = 3. The translated vertex is located at the point (-7, 3).

Reflection: If a is positive, the graph opens up. If a is negative, the graph is reflected over the x-axis.

Vertical Stretch/Compression: The value of a is not in the parenthesis: a(x)2. If |a| > 1, the graph stretches vertically away from the x-axis. If 0 < |a| < 1, the graph compresses vertically toward the x-axis. f(x) = 2x2 , a = 2, stretch vertically by factor of 2.

 

 

 

Horizontal and Vertical Stretch/Compression: Create a table of values of a horizontal and vertical stretch and compression.

Vertical Stretch: f(x) = 2x2 Hor. Compress: f(x) = (2x)2 1 2(1)2 = 2 2 2(2)2 = 8 3 2(3)2 = 18 x f(x) 1 2 3 a = 2   x f(x) 1 2 3 Hor. Compress: f(x) = (2x)2 x f(x) 1 (2∙1)2 =4 2 (2∙2)2 = 16 3 (2∙3)2 =81    

HW pg. 320 #’s 23-28, 31, 33-41, 45