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Math  /  Algebra

Questionparent punction: y=x3y=x^{3} find the equation with (−4,−1),(−5,−2),(−6,−3)(-4,-1),(-5,-2),(-6,-3) Descrive the trans formation that were applied to the parent function

Studdy Solution

STEP 1

1. The parent function is y=x3 y = x^3 .
2. We need to find a transformed function that passes through the given points (−4,−1)(-4,-1), (−5,−2)(-5,-2), and (−6,−3)(-6,-3).
3. Transformations can include translations, stretches, compressions, and reflections.

STEP 2

1. Identify the type of transformation applied to the parent function.
2. Determine the vertical translation.
3. Determine the horizontal translation.
4. Write the equation of the transformed function.
5. Verify the equation with the given points.

STEP 3

Identify the type of transformation applied to the parent function y=x3 y = x^3 .
The given points suggest that the function has been translated. We will check for vertical and horizontal translations.

STEP 4

Determine the vertical translation.
The parent function y=x3 y = x^3 has a point at (0,0) (0, 0) . The given points suggest a vertical translation. Let's calculate the vertical shift using one of the points, say (−4,−1)(-4, -1).
The vertical shift is the difference in the y-values:
Δy=−1−0=−1 \Delta y = -1 - 0 = -1
This suggests a vertical translation downward by 1 unit.

STEP 5

Determine the horizontal translation.
The parent function y=x3 y = x^3 has a point at (0,0) (0, 0) . The given points suggest a horizontal translation. Let's calculate the horizontal shift using the same point (−4,−1)(-4, -1).
The horizontal shift is the difference in the x-values:
Δx=−4−0=−4 \Delta x = -4 - 0 = -4
This suggests a horizontal translation to the left by 4 units.

STEP 6

Write the equation of the transformed function.
Considering the transformations, the equation of the transformed function is:
y=(x+4)3−1 y = (x + 4)^3 - 1
This accounts for a horizontal shift left by 4 units and a vertical shift down by 1 unit.

STEP 7

Verify the equation with the given points.
Check if the equation y=(x+4)3−1 y = (x + 4)^3 - 1 passes through the points (−4,−1)(-4, -1), (−5,−2)(-5, -2), and (−6,−3)(-6, -3).
1. For (−4,−1)(-4, -1): y=((−4)+4)3−1=03−1=−1 y = ((-4) + 4)^3 - 1 = 0^3 - 1 = -1 Correct.
2. For (−5,−2)(-5, -2): y=((−5)+4)3−1=(−1)3−1=−1−1=−2 y = ((-5) + 4)^3 - 1 = (-1)^3 - 1 = -1 - 1 = -2 Correct.
3. For (−6,−3)(-6, -3): y=((−6)+4)3−1=(−2)3−1=−8−1=−3 y = ((-6) + 4)^3 - 1 = (-2)^3 - 1 = -8 - 1 = -3 Correct.
The equation y=(x+4)3−1 y = (x + 4)^3 - 1 is verified.
The equation of the transformed function is y=(x+4)3−1 y = (x + 4)^3 - 1 .

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