QUADRATIC FUNCTION

Intro…

• Functions with the form

y=ax2+bx+c

are called quadratic functions and their graphs have a parabolic shape

• When we solve ax2+bx+c=0 we look for values of x that are x-intercepts (because we have y=0)

• The x-intercepts are called the solutions or roots of a quadratic equation

Solving Quadratic Equations by Graphing

• Quadratic equation

y=ax2+bx+c• ax2 is the quadratic term, bx is the linear

term, and c is the constant term

• A quadratic equation can have– two real solutions, – one real solution, – or no real solutions

Solving Quadratic Equations by Factoring

• Factor with the zero product property: if a*b=0 then either a=0 or b=0 or both are equal to 0

• Factoring by guess and check is useful, but you may have to try several combinations before you find the correct one

• While doing word problems examine your solutions carefully to make sure it is a reasonable answer

The Quadratic Formula and the Discriminant

• The quadratic formula gives the solutions of ax2 + bx + c = 0 when it is not easy to factor the quadratic or complete the square

• Quadratic formula:

• The b2 – 4ac term is called the discriminant and it helps to determine how many and what kind of roots you see in the solution

a

acbbx

2

4/ 2

Example

Graph y= -x2 - 2x + 8 and find its roots.

Vertex: (-1, 9)

Roots: (-4, 0) (2, 0)

Viewing window:

Xmin= -10

Xmax=10

Ymin= -10

Ymax= 10

POSSIBLE SHAPES

4 langkah menggambar kurva

• Step 1

Determine the basic shape. The graph has a U shape if a > 0, and an inverted U shape if a < 0.

• Step 2

Determine the y intercept. This is obtained by substituting x = 0 into the function, which gives y = c.

• Step 3

Determine the x intercepts (if any). These are obtained by solving the quadratic equation

• Step 4

Determine the vertex by finding the symmetry and substitute the value of the x symemtry

• The axis of symmetry is a line that divides a parabola into two equal parts that would match exactly if folded over on each other

• The vertex is where the axis of symmetry meets the parabola

• The roots or zeros (or solutions) are found by solving the quadratic equation for y=0 or looking at the graph

example

• F(x) = -x2 + 8 x – 12• Gambar grafiknya: 4 langkah.• 1. menentukan basic shape. Karena a < 0

maka INVERTED U SHAPE• 2. intercept dg sumbu y (x = 0) maka y = -12.

jadi grafik akan memotong y pada (0, -12)• 3. selesaikan persamaan tsb / cari nilai x nya• 4. cari sumbu tengahnya dan titik puncaknya

• The axis of symmetry is a line that divides a parabola into two equal parts that would match exactly if folded over on each other

• The vertex is where the axis of symmetry meets the parabola

• The roots or zeros (or solutions) are found by solving the quadratic equation for y=0 or looking at the graph

Graph with definitions shown: Three outcomes for number of roots:

One root:Two roots

No roots:

Vertex (2., -5.)

Root Root

Axis of Symmetry

-10 -8 -6 -4 -2 2 4 6 8 10

-10

-8

-6

-4

-2

2

4

6

8

10

-10 -8 -6 -4 -2 2 4 6 8 10

-10

-8

-6

-4

-2

2

4

6

8

10

-10 -9 -8 -7 -6 -5 -4 -3 -2 -1 1 2 3 4 5 6 7 8 9 10

1

2

3

4

5

6

7

8

9

10

Example

-x2: quadratic term

-2x: linear term

8: constant term

Vertex:

x=(-b/2a)

x= -(-2/2(-1))

x= 2/(-2)

x= -1

Solve for y:

y= -x2 -2x + 8

y= -(-1)2 -(2)(-1) + 8

y= -(1) + 2 + 8

y= 9 Vertex is (-1, 9)

For y= -x2 -2x + 8 identify each term, graph the equation, find the vertex, and find the solutions of the equation.

-10 -8 -6 -4 -2 2 4 6 8 10

-10

-8

-6

-4

-2

2

4

6

8

10

Root (-4., 0.) Root (2., 0.)

Vertex (-1., 9.)

end