Assignment 2

Parabolic Transformations

Forms of Parabolic Equations

 Standard form y = ax² + bx + c a > 0, parabola opens up a < 0, parabola opens down axis of symmetry: x = -b/2a Vertex form y = a(x - h)² + k (h, k) is the vertex a > 0, parabola opens up a < 0, parabola opens down | a | > 1, graph narrows | a | < 1, graph widenss Intercept form y = a(x - p)( x - q) x interepts are at p and q axis of symmetry between (p,0) and (q,0)

 From vertex to standard form expand and simplify the binomial From standard to vertex form complete the square Conclude

Graph the parabola y = 2x² + 3x - 4

h and k can be calculated using the above formulas giving the vertex form of the equation as

y = 2(x +.75)² - 5.125

The following transformations to the original graph occur.

 a = 2 h = -.75 k = -5.125

Graph the parabola y = 2x² + 3x - 4

In vertex form this is y = 2(x +.75)² - 5.125

Overlay new graph replacing each x by (x - 4)

Using standard form y = 2(x - 4)² + 3(x + 4)- 4

Simplifying gives y = 2x² - 13x + 16

In vertex form this is y = 2(x - 3.25)² - 5.125

or

Using vertex form y = 2((x - 4) +.75)² - 5.125

Simplifying gives y = 2x² - 13x + 16

Change the equation to move the vertex of the graph into the second quadrant

Want vertex at (-.75, 5.125)

so y = 2(x + .75)² + 5.125

Expand & Simplify y = 2x² + 3x + 6.25

Change the equation to produce a graph concave down

Vertex stays at (-.75, 5.125).

a becomes negative. (-1)(2) = -2

so y = -2(x + .75)² + 5.125

Expand & Simplify y = -2x² - 3x + 4