Graphing quadratic-equations-4818

Find vertex, roots and sketch parabolas

2. -
b
a2
•The standard form of a quadratic equation is y = ax
2
+ bx + c.
•The graph of a quadratic equation is a parabola.
•When a is positive, the graph opens up.
•When a is negative, the graph opens down.
•Every parabola has a vertex. For graphs opening up, the vertex is
a minimum (low point). For graphs opening down, the vertex is a
maximum (high point).
•The x-coordinate of the vertex is equal to .

3. To find the x coordinate of the vertex, use the equation
Then substitute the value of x back into the equation of the
parabola and solve for y.
x
b
a
=-
2
You are given the equation y=-x
2
+ 4x –1. Find the coordinates
of the vertex.
x
b
a
=-
2
x=2
-= =a b1 4,
(-
x=-
4
21)()
2
yx x=-+-41
y=-+-481
y=3
y=- + -
2
2421()()
The coordinates of the vertex are (2,3)

4. Choose two values of x that are to the right or left of the x-
coordinate of the vertex.
Substitute those values in the equation and solve for y.
Graph the points.
Since a parabola is symmetric, you can plot those same y-values on
the other side of the parabola, the same distance away from the
vertex.
xy = -x
2
+ 4x -1 y
1
-1
y = -(1)
2
+ 4(1) - 1
y = -1 +4 - 1
2
y = -(-1)
2
+ 4(-1) -1
y = -1 -4 -1
-6

5. x
y
Plot the vertex and the points from
your table of values: (2,3), (1,2),
(-1,-6).
Since a parabola is symmetric and
has a vertical line of symmetry
through the vertex of the parabola,
points which are the same distance
to the right of the vertex will also
have the same y-coordinate as
those to the left of the vertex.
Therefore, we can plot points at
(3,2) and (5,-6).
Sketch in the parabola.

6. Find the vertex of the following quadratic equations. Make a
table of values and graph the parabola.
1 4
2
. y x x= -
2 2 3
2
. y x=- +
3 6 4
2
. y =x x- +

7. y x x= -
2
4
a b= =-1 4
x
b
a
x
x
=-
=-
-
=
2
4
21
2
()
y x x
y
y
y
= -
= -
= -
=-
2
2
4
2 42
4 8
4
()
The vertex is at (2,-4)
x y = x
2
- 4x y
1 y = 1
2
- 4 ( 1 ) - 3
0 y = 0
2
- 4 ( 0 ) 0
x
y

8. y x=- +2 3
2
a b=- =2 0
x
b
a
x
x
=-
=-
-
=
2
0
2 2
0
( )
y x
y
y
=- +
=- +
=
2 3
20 3
3
2
2
()
The vertex is at (0,3)
x y = - 2x
2
+ 3 y
- 1 y = - 2 ( - 1 )
2
+ 3 1
- 2 y = - 2 ( - 2 )
2
+ 3 - 5
x
y

9. y x x= - +
2
6 4
a b= =-1 6
x
b
a
x
x
=-
=-
-
=
2
6
21
3
()
y x x
y
y
= - +
= - +
=-
2
2
6 4
3 63 4
5
()
The vertex is at (3,-5)
x y = x
2
- 6 x + 4 y
1 y = 1
2
- 6 ( 1 ) + 4 - 1
0 y = 0
2
- 6 ( 0 ) + 4 4
x
y

10. y x x= - +
2
6 4
a b= =-1 6
x
b
a
x
x
=-
=-
-
=
2
6
21
3
()
y x x
y
y
= - +
= - +
=-
2
2
6 4
3 63 4
5
()
The vertex is at (3,-5)
x y = x
2
- 6 x + 4 y
1 y = 1
2
- 6 ( 1 ) + 4 - 1
0 y = 0
2
- 6 ( 0 ) + 4 4
x
y