The Shifted Form Of A Parabola Homework

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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 + bx + c, where a, b, and c are constants. This is called the standard form of a parabola.

Sometimes, we want to shift or translate a parabola to a different position on the coordinate plane. For example, we might want to move it up or down, left or right, or both. To do this, we can use the shifted form of a parabola, which is y = a(x - h) + k, where h and k are constants that represent the horizontal and vertical shifts.

The shifted form of a parabola tells us how to move the vertex (the highest or lowest point) of the parabola from its original position at (0, c) to a new position at (h, k). To find the new vertex, we use these rules:

If h is positive, we move the vertex h units to the right.

If h is negative, we move the vertex h units to the left.

If k is positive, we move the vertex k units up.

If k is negative, we move the vertex k units down.

For example, if we have the parabola y = x - 4x + 5 in standard form, and we want to shift it 3 units to the right and 2 units down, we can use the shifted form of a parabola to write y = (x - 3) - 1. The new vertex is at (3, -1), which is 3 units to the right and 2 units down from the original vertex at (2, 1).

In this homework assignment, you will practice finding the shifted form of a parabola given its standard form and its new vertex coordinates. You will also practice finding the new vertex coordinates given the shifted form of a parabola. You can use this online calculator[^1^] to check your answers. Good luck!

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Let's look at an example of how to find the shifted form of a parabola given its standard form and its new vertex coordinates. Suppose we have the parabola y = -2x + 8x - 3 in standard form, and we want to shift it to have a new vertex at (-1, 7). How can we write the shifted form of the parabola

To find the shifted form of the parabola, we need to find the values of h and k that correspond to the new vertex coordinates. We can use these formulas:

h = -b/2a

k = c - b/4a

where a, b, and c are the constants in the standard form of the parabola. In this case, a = -2, b = 8, and c = -3. Plugging these values into the formulas, we get:

h = -8/2(-2) = 2

k = -3 - 8/4(-2) = 5

These are the original vertex coordinates of the parabola. To find the shifted vertex coordinates, we need to compare them with the new vertex coordinates. We can use these rules:

If the new x-coordinate is smaller than the original x-coordinate, we subtract the difference from h.

If the new x-coordinate is larger than the original x-coordinate, we add the difference to h.

If the new y-coordinate is smaller than the original y-coordinate, we subtract the difference from k.

If the new y-coordinate is larger than the original y-coordinate, we add the difference to k.

In this case, the new vertex coordinates are (-1, 7). Comparing them with the original vertex coordinates (2, 5), we see that:

The new x-coordinate is smaller than the original x-coordinate by 3 units, so we subtract 3 from h: h - 3 = 2 - 3 = -1.

The new y-coordinate is larger than the original y-coordinate by 2 units, so we add 2 to k: k + 2 = 5 + 2 = 7.

Therefore, the shifted form of the parabola is y = -2(x - (-1)) + 7. We can simplify this by removing the double negative sign: y = -2(x + 1) + 7. This is our final answer.

Now let's look at an example of how to find the new vertex coordinates given the shifted form of a parabola. Suppose we have the parabola y = (x + 4) - 9 in shifted form. What are the new vertex coordinates of this parabola

To find the new vertex coordinates of this parabola, we need to identify the values of h and k in the shifted form of the parabola. We can use these rules:

If there is a minus sign before x in parentheses, then h is positive and equal to that number.

If there is a plus sign before x in parentheses, then h is negative and equal to that number with a minus sign.

The value of k is equal to whatever number is outside of parentheses.

In this case, there is a plus sign before x in parentheses, so h is negative and equal to -4. The value of k is equal to -9. Therefore, the new vertex coordinates are (-4, -9).

We hope this article has helped you understand how to find and use the shifted form of a parabola for your homework. Remember that you can always check your answers with an online calculator or ask your teacher for help if you get stuck. Happy learning!

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