## Course 2, Unit 4 - Power Models

Summary
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.

Key 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 below.