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