1. Lesson 1-7 `Quadratic Functions and their Graphs Object
Lesson 1-7 `Quadratic Functions and their Graphs Object. To define and graph quadratic functions .
Axis
If a graph has an axis of symmetry when fold the graph along the axis the two halves coincide
Vertex
Y = ax2+bx + c
a> 0

2. The vertex of the parabola is the point where the axis of symmetry intersects the parabola.
Y = -3x2
The bigger |a | is the narrower the parabola becomes, the smaller |a| becomes the wider the parabola becomes.
The y intercept of y = ax2+ bx +c
Is the value of c
y = .25x2
Y = ax2+bx+c
a < 0
Y = ax2+bx+c
a > 0

3. The x – intercepts are the real roots of ax2+bx+c =0
The x – intercepts are the real roots of ax2+bx+c =0. Since the quadratic equation may have two, one,or no real roots. Depending on the value of the discriminant b2 – 4ac we have three possibilities as shown below
If b2 – 4ac <0 there are no
x –intercepts
(0,c)
(0,c)
If b2 – 4ac >0
There are 2
X- intercepts
If b2 – 4ac =0 there is only one x –intercept where the vertex and the x axis meet.

4. The axis of symmetry is a vertical line halfway between the x-intercepts. It passes through the vertex.The equation of the axis is x = - b 2a
Example1: Sketch the graph of y=2x2 - 8x + 5. Label the intercepts, axis of symmetry, and the vertex.
Step 1: to find the vertex let x =0
y = 2(0)2 -8(0) +5 = y= So y = 5
Step 2: To find the x- intercepts solve
2x2 -8x+5 = 0 A=2 , B= -8, C = 5

5. X X Step 2: To find the x- intercepts solve
2x2 -8x+5 = 0 A=2 , B= -8, C = 5
Use X X X

6. The axis of symmetry is the line x = - b = - ( -8 ) = - -8 = 2 2a 2(2) 4
Step 4:Since the vertex is on the axis of symmetry its x value is 2. To find the y value substitute the x with 2
Y = 2(2)2-8(2)+5
Y= 2(4) – = 8+5 – 16 = 13 – 16 y =-3
The vertex (2,-3)
(0,5)
(4,5)
(3,2)
(.8,0)
If the equation is written in the form of
y = a(x –h)2 + k
Then the vertex is (h,k)
The vertex of the parabola y = 2(x – 3) is (3,7)
The vertex of the parabola y = -4(x – 9) is (-9,2)
(2,-3)

7. Example .2: Find the vertex of the parabola y = -2x2 +12x + 4 by completing
The square.
b. Find the x and y intercepts
y = -2x2 +12x + 4
= -2(x2 - 6x ) + 4
= -2(x2 - 6x +9) + 4 – (-2)(9)
= -2 (x -3)2 + 22
The vertex is (3,22)
b.) When x = 0 then y = 4 thus the y intercept is 4 which is the constant term of the equation.
To find the x intercept let y = 0 Using the completed square form of the equation
-2 (x -3) = 0
-2 (x -3)2 = -----Divide by -2
(x -3)2 = 11 ------take the square root of both sides

8. Sketching the Graph of y = ax2 + bx + c , a ≠ 0
Get a quick mental picture of the graph as follows
a.) if a > 0 the graph opens upwards
if a < 0 the graph opens downwards
b.) if b2 – 4ac > 0 the graph has 2 intercepts
If b2 – 4ac = 0 the graph has one intercept
if b2 – 4ac < 0 the graph has no intercepts
2. Find the vertex and axis of the parabola as follows
Method 1: The equation of the axis is x = - b Substitute the value of x into 2a
y = ax2+bx+c to find the y coordinate of the vertex.
Method 2: Rewrite the equation in the form of
y = a(x – h)2 + k
The vertex is (h,k) and the axis is x = h
3. Find the x and y intercepts
a.) Using y = ax2+bx+c , the y intercept is always c
b.) To find the x intercept solve ax2+bx+c =0 (from using Method 1)
or solve a(x – h)2 + k =0 (from using Method 2)

9. Example 3: Where does the line y = 2x + 5 intersect the parabola 8 – x2
Set 2x + 5 equal to 8 – x2 and solve for x
2x + 5 = 8 – x2
x x = 0
(x + 3)(x – 1) = 0
x + 3 = 0 or x – 1 = 0
x = x = 1
Substitute x = -3 into the equation y = 2x + 5 to get y = -1 and 1 into the equation
to get y = 7. The intersection points are (-3,-1) and (1,7)
Use a computer or graphing calculator to graph the equation y=2x + 5 and 8 – x2

10. Example 4: Find an equation of the function whose graph is a parabola with x-intercepts 1 and 4 and y- intercept -8
Any parabola with x-intercepts 1 and 4 has an equation of the form y=a(x-1)(x-4) for some constant a. We use the fact that the parabola contains (0,-8) to find the value of a. -8 = A(0 – 1)(0 – 4) = (-1)(-4) -8 = 4A -2 = A Then the equation is y = -2(x -1)(x-4) or y = -2x2+10x -8

11. Homework pg. 41

12. Videos http://www.youtube.com/watch?v=c74m88mbi9k