b If the parabola opens upwards, the vertex is located at the minimum point of the parabola. Otherwise, it is located at the maximum point of the parabola.
C
c Make a table of values for the functions.
D
d If the parabola opens upwards, the vertex is located at the minimum point of the parabola. Otherwise, it is located at the maximum point of the parabola.
E
e What is the difference between the graphs?
A
aGraph:
B
bVertex: (0,0)
Example Points: (- 2,4) and (2,4)
C
cGraph:
D
dTable:
Function
Vertex
Points
y=x^2+2
(0,2)
(- 2,6) and (- 2,6)
y=x^2+4
(0,4)
(- 2,8) and (- 2,8)
y=x^2+6
(0,6)
(- 2,10) and (- 2,10)
E
e See solution.
Practice makes perfect
a Let's make a table of values to graph the function y=x^2.
x
x^2
y=x^2
- 2
( - 2)^2
4
- 1
( - 1)^2
1
0
( 0)^2
0
1
( 1)^2
1
2
( 2)^2
4
Let's plot the points ( - 2, 4), ( - 1, 1), ( 0, 0), ( 1, 1), and ( 2, 4) then connect them with a smooth curve.
b Since the parabola opens upward, the vertex is located at the minimum point of the parabola. It is the point (0,0).
The points (- 2,4) and (2,4) lie on the parabola.
c To graph the given functions, we will make a table of values of the functions.
x
- 2
- 1
0
1
2
y=x^2+2
( - 2)^2+2= 6
( - 1)^2+2= 3
( 0)^2+2= 2
( 1)^2+2= 3
( 2)^2+2= 6
y=x^2+4
( - 2)^2+4= 8
( - 1)^2+4= 5
( 0)^2+4= 4
( 1)^2+4= 5
( 2)^2+4= 8
y=x^2+6
( - 2)^2+6= 10
( - 1)^2+6= 7
( 0)^2+6= 6
( 1)^2+6= 7
( 2)^2+6= 10
The ordered pairs ( - 2, 6), ( - 1, 3), ( 0, 2), ( 1, 3), and ( 2, 6) all lie on the function y=x^2+2.
The ordered pairs ( - 2, 8), ( - 1, 5), ( 0, 4), ( 1, 5), and ( 2, 8) all lie on the function y=x^2+4.
The ordered pairs ( - 2, 10), ( - 1, 7), ( 0, 6), ( 1, 7), and ( 2, 10) all lie on the function y=x^2+6.
We will plot and connect these points with a smooth curve.
d Since each parabola opens upward, the vertices are located at the minimum point of the graphs.
Let's show them in a table.
Function
Vertex
Points
y=x^2+2
(0, 2)
(- 2,6) and (- 2,6)
y=x^2+4
(0, 4)
(- 2,8) and (- 2,8)
y=x^2+6
(0,6)
(- 2,10) and (- 2,10)
e We can conclude that when we add a positive number, k, to the function y=x^2, the vertex increase by k units.
y=x^2+k
Moreover, the graph of the function is translated k units up.
