4.2 Graph Quadratic Functions in Vertex or Intercept Form

Vertex form

A quadratic function written in the form y = a(x - h)² + k

Intercept form

A quadratic function written in the form y = a(x - p)(x - q)

GRAPH OF VERTEX FORM y = a(x - h)² + k

The graph of y = a(x - h)² + k is the parabola y = ax² translated _horizontally_ h units and _vertically_ k units.

· The vertex is (_h_, _k_ ).

· The axis of symmetry is x = _h_.

· The graph opens up if a _<_ 0 and down if a _<_ 0.

Example 1

Graph a quadratic function in vertex form

Graph y = (x + l)² - 2.

1. Identify the constants a =_{____} , h = _-1_ and k = _-2_.

Because a > 0, the parabola opens _up_.

2. Plot the vertex (h, k) = ( _-1_, _-2_ )and draw the axis of symmetry at x = _-1_.

3. Evaluate the function for two values of x.

x = 1; y = 0

x = 3; y = 6

Plot the points (1, _0_ ) and (3, _6_ ) and their reflections in the axis of symmetry.

4. Draw a parabola through the plotted points.

GRAPH OF INTERCEPT FORM y = a(x - p)(x - q):

Characteristics of the graph y = a(x - p)(x - q):

· The x-intercepts are _p_ and _q_.

· The axis of symmetry is halfway between ( _p , 0) and ( _q_ , 0). It has

equation x =

· The graph opens up if a _>_ 0 and opens down if a _<_ 0.

Example 2

Graph a quadratic function in intercept form

Graph y = -2(x - 1)(x - 5).

1. Identify the x-intercepts. Because p = _1_ and q = _5_, the x-intercepts occur at the points ( _1_, 0) and (_5_, 0).

2. Find the coordinates of the vertex.

x = = = _3_

y = _-2(3 - 1)(3 - 5)_ = _8_

So, the vertex is (_3_, _8_).

3. Draw a parabola through the vertex and the points where the x-intercepts occur.

FOIL METHOD

Words To multiply two expressions that each contain two terms, add the products of the _First_ terms, the _Outer_ terms, the _Inner_ terms, and the _Last_ terms.

Example F O I L

(x + 4)(x + 7) = x² + 7x + 4x + 28 = x² + 11x + 28

Example 3

Change from intercept form to standard form

Write y = 3(x + 2)(x - 5) in standard form.

y = 3(x + 2)(x - 5)		Original function
	= 3_(x^2 - 5x + 2x - 10)_	Multiply using FOIL.
	= 3_(x^2 - 3x - 10)_	Combine like terms.
	= _3x^2 - 9 - 30_	Distributive property

Example 4

Change from vertex form to standard form

Write f(x) = -5(x + 2)² + 8 in standard form.

f(x) = -5(x + 2)^2 + 8		Original function
	= -5( _x+2_ )( _x+2_ ) + 8	Rewrite (x + 2)^2
	= -5(x^2 + 2x + 2x +4 ) + 8	Multiply using FOIL.
	= -5( _x^2 + 4x + 4 ) + 8	Combine like terms.
	= _-5x^2 - 20x - 20x_ + 8	Distributive property
	= _-5x^2 - 20x - 12_	Combine like terms.