2, Unit 4 - Power Models
In the "Power Models" unit, students learn to recognize types of situations
which can be represented by power models, inverse power models, and quadratic
models, and to compare patterns found in the tables and graphs of models
of the form y = ax2, y = ax3, y = a/x2, y = a/x3,
and y = ax2 + bx + c. In
addition, students begin to develop an understanding of rate of change
in the context of these models. Some specific skills are developed here:
solving quadratics by graphical methods and simplifying exponential and
radical expressions. In addition, the vocabulary and concepts of direct
and inverse variation are introduced.
Ideas from Course 2, Unit 4
- Power model: a function with equation of the form y = axn or a/xn.
- Direct variation power model: a function with equation y = axn (n > 0),
for example, the relationship between volume of a cube and edge length
is modeled by a direct variation function. We say that the volume varies
directly with the cube of the edge length, because volume is given
by the equation v = s3. Thus, as s3 increases, v increases
with it. This is a cubic, or power 3, model.
- Inverse variation power model: a function with equation y = a/xn (n > 0).
For example, the relationship between time and speed, for a fixed distance
traveled, is modeled by an inverse variation function. We say that
time varies inversely with speed, because the time is given by the
equation t = d/s, where d is the fixed distance.
Thus, as s increases, t decreases. For example, if the
fixed distance is 300 miles then traveling at 20 mph will
lead to a trip time of 15 hours. Increasing speed to 50 mph
will cause the time to decrease to 6 hours.
- Quadratic model: a function with equation in the form y = ax2 + bx + c. The
relationship between height (in feet) of a kicked ball and its time
in flight (in seconds) is modeled reasonably well by a quadratic function.
For example, if h = -16t2 + 50t + 3,
the ball's height in feet after t seconds depends on the initial height
(3 feet in this example), the initial velocity of the ball (50 ft/sec
in this example), and the effect of gravity (indicated by the -16 ft/sec
in this example).
- Radical or fractional power model: a function with equation
in the form y = xn where n is a
fraction greater than 0. For example, the relationship between the
length of a side of a square and the area of the square is given by S = or .
- Direct variation power model graph: passes through the origin.
If the power is odd, then the end behavior is in opposite directions,
and the curve has rotational symmetry around the origin (the
point where the x- and y-axes intersect). If the power
is even, then the end behavior is in the same direction, and the curve
has reflection symmetry across the y-axis.
- Inverse variation power model graph: does not cross the axes;
the axes are asymptotes. As with direct variation power models, the
odd powers are symmetric about the origin. The even powers are symmetric
across the y-axis (a > 0).
- Quadratic function graph: is a parabola. It will have a maximum
or a minimum value, and be symmetric about a vertical axis. The
rate of change is not constant.
- Radical or fractional power model graph: resembles the ones