1.2 Graphing Quadratic 1.2 Graphing Quadratic Functions In Vertex or Functions In Vertex or

Intercept FormIntercept Form

• DefinitionsDefinitions

• 2 more forms for a quad. function2 more forms for a quad. function

• Steps for graphing Vertex and Intercept Steps for graphing Vertex and Intercept

• ExamplesExamples

• Changing between eqn. formsChanging between eqn. forms

Vertex Form EquationVertex Form Equationy=a(x-h)y=a(x-h)22+k+k

• If a is positive, parabola opens upIf a is positive, parabola opens up

If a is negative, parabola opens down.If a is negative, parabola opens down.

• The vertex is the point (h,k).The vertex is the point (h,k).

• The axis of symmetry is the vertical The axis of symmetry is the vertical line x=h.line x=h.

• Don’t forget about 1 point on either Don’t forget about 1 point on either side of the vertex! (3 points total!)side of the vertex! (3 points total!)

Example 2: GraphExample 2: Graphy=-½(x+3)y=-½(x+3)22+4 y=a(x-h)+4 y=a(x-h)22+k+k

• a is negative (a = -½), so parabola opens down.a is negative (a = -½), so parabola opens down.• Vertex is (h,k) or (-3,4)Vertex is (h,k) or (-3,4)• Axis of symmetry is the vertical line x = -3Axis of symmetry is the vertical line x = -3• Table of values Table of values x y x y

-1 2-1 2 -2 3.5 -2 3.5

-3 4-3 4 -4 3.5-4 3.5 -5 2-5 2

Vertex (-3,4)

(-4,3.5)

(-5,2)

(-2,3.5)

(-1,2)

x=-3

Now you try one!Now you try one!y=a(x-h)y=a(x-h)22+k+k

y=2(x-1)y=2(x-1)22+3+3

•Open up or down?Open up or down?

•Vertex?Vertex?

•Axis of symmetry?Axis of symmetry?

•Table of values with 3 points?Table of values with 3 points?

(-1, 11)

(0,5)

(1,3)

(2,5)

(3,11)

X = 1

Civil EngineeringCivil 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 ?

SOLUTIONSOLUTION

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 EquationIntercept Form Equationy=a(x-p)(x-q)y=a(x-p)(x-q)

• The x-intercepts are the points (p,0) and The x-intercepts are the points (p,0) and (q,0).(q,0).

• The axis of symmetry is the vertical line x=The axis of symmetry is the vertical line x=

• The x-coordinate of the vertex isThe x-coordinate of the vertex is

• To find the y-coordinate of the vertex, plug To find the y-coordinate of the vertex, plug the x-coord. into the equation and solve for y.the x-coord. into the equation and solve for y.

• If a is positive, parabola opens upIf a is positive, parabola opens up

If a is negative, parabola opens down.If a is negative, parabola opens down.

2

qp 2

qp

Example 3: Graph y=-(x+2)(x-Example 3: Graph y=-(x+2)(x-4)4)

y=a(x-p)(x-q)y=a(x-p)(x-q)• Since a is negative, Since a is negative,

parabola opens parabola opens down.down.

• The x-intercepts are The x-intercepts are (-2,0) and (4,0)(-2,0) and (4,0)

• To find the x-coord. To find the x-coord. of the vertex, useof the vertex, use

• To find the y-coord., To find the y-coord., plug 1 in for x. plug 1 in for x.

• Vertex (1,9)Vertex (1,9)

2

qp

12

2

2

42

x

9)3)(3()41)(21( y

•The axis of The axis of symmetry is the symmetry is the vertical line x=1 vertical line x=1 (from the x-coord. (from the x-coord. of the vertex)of the vertex)

x=1

(-2,0) (4,0)

(1,9)

Now you try one!Now you try one!y=a(x-p)(x-q)y=a(x-p)(x-q)

y=2(x-3)(x+1)y=2(x-3)(x+1)

•Open up or down?Open up or down?

•X-intercepts?X-intercepts?

•Vertex?Vertex?

•Axis of symmetry?Axis of symmetry?

(-1,0) (3,0)

(1,-8)

x=1

FootballFootball

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 ?

SOLUTIONSOLUTIONa. 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)?SOLUTIONa. 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 15.625 yards.

Changing from vertex or Changing from vertex or intercepts form to standard intercepts form to standard

formform• The key is to FOIL! (first, outside, inside, The key is to FOIL! (first, outside, inside,

last)last)

• Ex: y=-(x+4)(x-9)Ex: y=-(x+4)(x-9) Ex: y=3(x-1)Ex: y=3(x-1)22+8+8

=-(x=-(x22-9x+4x-36)-9x+4x-36) =3(x-1)(x-1)+8 =3(x-1)(x-1)+8

=-(x=-(x22-5x-36)-5x-36) =3(x =3(x22-x--x-x+1)+8x+1)+8

y=-xy=-x22+5x+36+5x+36 =3(x =3(x22--2x+1)+82x+1)+8

=3x=3x22-6x+3+8-6x+3+8

y=3xy=3x22-6x+11-6x+11

Change from intercept Change from intercept form to standard formform 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 Change from vertex form to standard formto 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= 4(x 2 – x – x + 1) + 9Multiply using FOIL.

Rewrite (x – 1)2.

= 4(x 2 – 2x + 1) + 9 Combine like terms.

= 4x 2 – 8x + 4 + 9 Distributive property

= 4x 2 – 8x + 13 Combine like terms.

Assignment

p. 15,

3-39 every 3rd problem

(3,6,9,12,…)