An electric device delivers a current of [tex]$15.0 A$[/tex] for 30 seconds. How many electrons flow through it?
Answer (2)
Vertical Shift: y = x 2 becomes y = x 2 + 3 (shifts upward).
Vertical Stretch/Compression: y = 2 1 x 2 becomes y = 3 x 2 (stretch).
Horizontal Shift: y = x 2 becomes y = ( x − 5 ) 2 (shifts right).
Reflection: y = 3 x 2 becomes y = − 3 x 2 (reflects across x-axis).
Explanation
Problem Analysis We are given pairs of quadratic functions and asked to compare and contrast them. We will identify the transformations that relate one function to the other, noting vertical shifts, vertical stretches/compressions, horizontal shifts, and reflections.
Comparison of Function Pair 34
y = x 2 , y = x 2 + 3 : The second function is a vertical shift of the first function by 3 units upward.
Comparison of Function Pair 35
y = 2 1 x 2 , y = 3 x 2 : The second function is a vertical stretch of the first function. The first function is vertically compressed compared to y = x 2 , while the second is stretched.
Comparison of Function Pair 36
y = x 2 , y = ( x − 5 ) 2 : The second function is a horizontal shift of the first function by 5 units to the right.
Comparison of Function Pair 37
y = 3 x 2 , y = − 3 x 2 : The second function is a reflection of the first function about the x-axis.
Comparison of Function Pair 38
y = x 2 , y = − 4 x 2 : The second function is a vertical stretch by a factor of 4 and a reflection about the x-axis.
Comparison of Function Pair 39
y = x 2 − 1 , y = x 2 + 2 : The second function is a vertical shift of the first function by 3 units upward.
Comparison of Function Pair 40
y = 2 1 x 2 + 3 , y = − 2 x 2 : The second function is a vertical stretch by a factor of 4, a reflection about the x-axis, and a vertical shift. The first function is vertically compressed and shifted up.
Comparison of Function Pair 41
y = x 2 − 4 , y = ( x − 4 ) 2 : The second function is a horizontal shift of the first function. The first function is shifted down, while the second is shifted to the right.
Final Summary In summary, we have identified the transformations relating each pair of quadratic functions. These transformations include vertical and horizontal shifts, vertical stretches and compressions, and reflections about the x-axis.
Examples
Understanding transformations of functions is crucial in many fields. For instance, in physics, understanding how graphs shift and stretch can help analyze the motion of objects under different forces. In economics, transformations can model how changes in tax rates affect supply and demand curves. In computer graphics, these transformations are fundamental for manipulating objects in 2D and 3D space. For example, the function y = x 2 can represent the trajectory of a projectile, and transformations like y = ( x − a ) 2 + b can model the same trajectory under different initial conditions or gravitational forces.
In a device with a current of 15.0 A flowing for 30 seconds, approximately 2.81 x 10^21 electrons pass through it. This calculation involves using the relationship between current, charge, and the charge of an electron. Therefore, the total charge is found first and then converted into the number of electrons.
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