Theshiftedformofaparabolahomework

The Shifted Form of a Parabola Homework

A parabola is a curve that has the shape of an arch. It can be described by the equation y = ax^2 + bx + c, where a, b, and c are constants. The graph of a parabola is symmetric about a vertical line called the axis of symmetry, which passes through the vertex, the highest or lowest point on the parabola.

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Sometimes, we may want to shift or transform a parabola to obtain a different parabola. For example, we may want to move the vertex to a different location, or change the width or direction of the parabola. To do this, we can use the shifted form of a parabola, which is given by the equation y = a(x - h)^2 + k, where a is the same as before, and h and k are constants that determine how much the parabola is shifted horizontally and vertically.

How to Shift a Parabola Horizontally

To shift a parabola horizontally, we need to change the value of h in the shifted form. The value of h tells us how much the parabola is moved to the right or left from its original position. If h is positive, then the parabola is shifted to the right by h units. If h is negative, then the parabola is shifted to the left by -h units. For example, if we have the parabola y = x^2, which has its vertex at (0, 0), and we want to shift it to the right by 3 units, we can use the equation y = (x - 3)^2, which has its vertex at (3, 0).

A graph showing two parabolas: y = x^2 in blue and y = (x - 3)^2 in red. The blue parabola has its vertex at (0, 0) and the red parabola has its vertex at (3, 0). The red parabola is shifted to the right by 3 units from the blue parabola.

How to Shift a Parabola Vertically

To shift a parabola vertically, we need to change the value of k in the shifted form. The value of k tells us how much the parabola is moved up or down from its original position. If k is positive, then the parabola is shifted up by k units. If k is negative, then the parabola is shifted down by -k units. For example, if we have the parabola y = x^2, which has its vertex at (0, 0), and we want to shift it up by 4 units, we can use the equation y = x^2 + 4, which has its vertex at (0, 4).

A graph showing two parabolas: y = x^2 in blue and y = x^2 + 4 in green. The blue parabola has its vertex at (0, 0) and the green parabola has its vertex at (0, 4). The green parabola is shifted up by 4 units from the blue parabola.

How to Combine Horizontal and Vertical Shifts

We can also combine horizontal and vertical shifts to move a parabola in any direction. To do this, we need to change both h and k in the shifted form. The values of h and k tell us the coordinates of the new vertex of the parabola. For example, if we have the parabola y = x^2, which has its vertex at (0, 0), and we want to shift it to the right by 3 units and up by 4 units, we can use the equation y = (x - 3)^2 + 4, which has its vertex at (3, 4).

A graph showing two parabolas: y = x^2 in blue and y = (x - 3)^2 + 4 in purple. The blue parabola has its vertex at (0, 0) and the purple parabola has its vertex at (3, 4). The purple parabola is shifted to the right by 3 units and up by 4 units from the blue parabola.

How to Practice Shifting Parabolas

One way to practice shifting parabolas is to use online resources that provide interactive exercises and feedback. For example, you can use [Khan Academy], which offers a series of videos and quizzes on shifting parabolas . You can also use [Desmos], which is a free online graphing calculator that allows you to explore how changing the values of a, h, and k affect the shape and position of a parabola. By using these tools, you can improve your understanding of the shifted form of a parabola and how to apply it to different situations.

Conclusion

The shifted form of a parabola is a useful way to describe how a parabola can be moved or transformed on a coordinate plane. By changing the values of h and k, we can shift a parabola horizontally and vertically, respectively. By combining these shifts, we can move a parabola in any direction. By practicing with online resources, we can master this skill and use it for various purposes.