In families of linear graphs, lines either have the same slope or the same y-intercept. However, in families of parabolas, graphs either share a vertex or an axis of symmetry, or both. Also, a family can consist of parabolas of the same shape.
Graph each group of equations on the same screen. Compare and
contrast the graphs. What conclusions can be drawn?
1.
y
= x2,
y = 0.2x2,
y = 3x2
Alternative Solutions:
Each graph opens upward and has its
vertex at the origin. Therefore, these equations are a family of parabolas. The
graph of y = 0.2x2 is wider than the graph of y =
x2. The graph of y = 3x2 is more
narrow than the graph of y = x2.
2.
y
= x2, y
= x2 – 6, y
= x2 + 3
Alternative Solutions:
Each
graph opens upward and has the same shape as y = x2 so
they form a family. Yet, each parabola has a different vertex located along the
y-axis. A constant greater than 0 shifts the graph upward, and a
constant less than 0 shifts the graph downward along the axis of symmetry.
3.
y = x2, y = (x +
2)2, y = (x – 4)2
Alternative Solutions:
Each graph
opens upward and has the same shape as y = x2. However,
each parabola has a different vertex located along the x-axis. Find the
number for x that results in 0 inside the parentheses. The graph shifts this
number of units to the left or right.
4.
y
= x2, y = (x – 7)2
+ 2
Alternative Solutions:
The graph
of y = (x – 7)2 + 2 has the same shape as the graph of
y = x2. However, it shifts to the right 7 units
because a positive 7 will result in zero inside the parentheses. It also shifts
upward 2 units because of the constant 2 outside the parentheses.
Sometimes computers are used to generate families of graphs.
Computer
Animation Link
5. In a computer game, a player dodges space shuttles that are shaped like
parabolas. Suppose the vertex of one shuttle is at the origin. The shuttle’s
initial shape and position are given by the equation y = 0.5x2.
It leaves the screen with its vertex at (6, 5). Find an equation to model the
final shape and position of the shuttle.
Alternative Solutions:
The
shape of the shuttle remains the same. However, the vertex shifts from (0, 0)
to the right 6 units and up 5 units.
Begin
with the original equation.
y = 0.5x2
y = 0.5(x – 6)2
If x = 6, then x
– 6 = 0. Shift the vertex to the right 6 units.
y = 0.5(x – 6)2 + 5
The 5 outside the parentheses shifts the entire parabola up 5 units.
So, the
final shape and position of the shuttle can be described by the equation y
= 0.5(x – 6)2 + 5.
Sumber
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