# 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.