1.2 Graphing Quadratic Functions In Vertex or Intercept Form Definitions 2 more forms for a quad. function Steps for graphing Vertex and Intercept Examples Changing between eqn. forms

Vertex Form Equation y=a(x-h)2+k If a is positive, parabola opens up If a is negative, parabola opens down. The vertex is the point (h,k). The axis of symmetry is the vertical line x=h. Don’t forget about 1 point on either side of the vertex! (3 points total!)

Example 2: Graph y=-½(x+3)2+4 y=a(x-h)2+k a is negative (a = -½), so parabola opens down. Vertex is (h,k) or (-3,4) Axis of symmetry is the vertical line x = -3 Table of values x y -1 2 -2 3.5 -3 4 -4 3.5 -5 2 Vertex (-3,4) (-4,3.5) (-2,3.5) (-5,2) (-1,2) x=-3

Now you try one! y=a(x-h)2+k y=2(x-1)2+3 Open up or down? Vertex? Axis of symmetry? Table of values with 3 points?

(-1, 11) (3,11) X = 1 (0,5) (2,5) (1,3)

Civil Engineering The Tacoma Narrows Bridge in Washington has two towers that each rise 307 feet above the roadway and are connected by suspension cables as shown. Each cable can be modeled by the function. y = (x – 1400)2 + 27 1 7000 where x and y are measured in feet. What is the distance d between the two towers ?

SOLUTION The vertex of the parabola is (1400, 27). So, a cable’s lowest point is 1400 feet from the left tower shown above. Because the heights of the two towers are the same, the symmetry of the parabola implies that the vertex is also 1400 feet from the right tower. So, the distance between the two towers is d = 2 (1400) = 2800 feet.

Intercept Form Equation y=a(x-p)(x-q) The x-intercepts are the points (p,0) and (q,0). The axis of symmetry is the vertical line x= The x-coordinate of the vertex is To find the y-coordinate of the vertex, plug the x-coord. into the equation and solve for y. If a is positive, parabola opens up If a is negative, parabola opens down.

Example 3: Graph y=-(x+2)(x-4) y=a(x-p)(x-q) Since a is negative, parabola opens down. The x-intercepts are (-2,0) and (4,0) To find the x-coord. of the vertex, use To find the y-coord., plug 1 in for x. Vertex (1,9) The axis of symmetry is the vertical line x=1 (from the x-coord. of the vertex) (1,9) (-2,0) (4,0) x=1

Now you try one! y=a(x-p)(x-q) y=2(x-3)(x+1) Open up or down? X-intercepts? Vertex? Axis of symmetry?

x=1 (-1,0) (3,0) (1,-8)

Football The path of a placekicked football can be modeled by the function y = – 0.026x(x – 46) where x is the horizontal distance (in yards) and y is the corresponding height (in yards). a. How far is the football kicked ? b. What is the football’s maximum height ?

SOLUTION a. Rewrite the function as y = – 0.026(x – 0)(x – 46). Because p = 0 and q = 46, you know the x - intercepts are 0 and 46. So, you can conclude that the football is kicked a distance of 46 yards. b. To find the football’s maximum height, calculate the coordinates of the vertex.    

WHAT IF? In Example 4, what is the maximum height of the football if the football’s path can be modeled by the function y = – 0.025x(x – 50)? SOLUTION a. Rewrite the function as y = – 0.025(x – 0) (x – 50). Because p = 0 and q = 50, you know the x - intercepts are 0 and 50. So, you can conclude that the football is kicked a distance of 50 yards. b. To find the football’s maximum height, calculate the coordinates of the vertex.    

The maximum height is the y-coordinate of the vertex, or about 15 The maximum height is the y-coordinate of the vertex, or about 15.625 yards.

Changing from vertex or intercepts form to standard form The key is to FOIL! (first, outside, inside, last) Ex: y=-(x+4)(x-9) Ex: y=3(x-1)2+8 =-(x2-9x+4x-36) =3(x-1)(x-1)+8 =-(x2-5x-36) =3(x2-x-x+1)+8 y=-x2+5x+36 =3(x2-2x+1)+8 =3x2-6x+3+8 y=3x2-6x+11

Change from intercept form to standard form Write y = – 2 (x + 5) (x – 8) in standard form. y = – 2 (x + 5) (x – 8) Write original function. = – 2 (x 2 – 8x + 5x – 40) Multiply using FOIL. = – 2 (x 2 – 3x – 40) Combine like terms. = – 2x 2 + 6x + 80 Distributive property

Change from vertex form to standard form Write f (x) = 4 (x – 1)2 + 9 in standard form. f (x) = 4(x – 1)2 + 9 Write original function. = 4(x – 1) (x – 1) + 9 Rewrite (x – 1)2. = 4(x 2 – x – x + 1) + 9 Multiply using FOIL. = 4(x 2 – 2x + 1) + 9 Combine like terms. Distributive property = 4x 2 – 8x + 4 + 9 = 4x 2 – 8x + 13 Combine like terms.

Assignment p. 15, 3-39 every 3rd problem (3,6,9,12,…)