Fill In The Table By Calculating The Value Of $y$ For Each Given $x$ Using The Equation $y=(x+2)^2(x-1)$.$[ \begin{array}{c|c} x & Y \ \hline -3 & \ -2 & \ -1 & \ 0 & \ 1 & \ 2 &

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Solving for y in the Given Equation

In this article, we will explore the process of solving for y in the given equation $y=(x+2)^2(x-1)$. We will use this equation to calculate the value of y for each given x and fill in the table accordingly.

Understanding the Equation

The given equation is a quadratic equation in the form of $y=(x+2)^2(x-1)$. This equation can be expanded to $y=(x^2+4x+4)(x-1)$, which simplifies to $y=x3+3x2-4x-4$. This equation represents a cubic function, which can have one or three real roots.

Calculating y for Each Given x

To calculate the value of y for each given x, we will substitute the value of x into the equation $y=x3+3x2-4x-4$. We will then simplify the expression to find the value of y.

Calculating y for x = -3

To calculate the value of y for x = -3, we will substitute x = -3 into the equation $y=x3+3x2-4x-4$. This gives us:

y=(−3)3+3(−3)2−4(−3)−4y=(-3)^3+3(-3)^2-4(-3)-4

Simplifying the expression, we get:

y=−27+27+12−4y=-27+27+12-4

y=−12y=-12

Calculating y for x = -2

To calculate the value of y for x = -2, we will substitute x = -2 into the equation $y=x3+3x2-4x-4$. This gives us:

y=(−2)3+3(−2)2−4(−2)−4y=(-2)^3+3(-2)^2-4(-2)-4

Simplifying the expression, we get:

y=−8+12+8−4y=-8+12+8-4

y=8y=8

Calculating y for x = -1

To calculate the value of y for x = -1, we will substitute x = -1 into the equation $y=x3+3x2-4x-4$. This gives us:

y=(−1)3+3(−1)2−4(−1)−4y=(-1)^3+3(-1)^2-4(-1)-4

Simplifying the expression, we get:

y=−1+3+4−4y=-1+3+4-4

y=2y=2

Calculating y for x = 0

To calculate the value of y for x = 0, we will substitute x = 0 into the equation $y=x3+3x2-4x-4$. This gives us:

y=(0)3+3(0)2−4(0)−4y=(0)^3+3(0)^2-4(0)-4

Simplifying the expression, we get:

y=−4y=-4

Calculating y for x = 1

To calculate the value of y for x = 1, we will substitute x = 1 into the equation $y=x3+3x2-4x-4$. This gives us:

y=(1)3+3(1)2−4(1)−4y=(1)^3+3(1)^2-4(1)-4

Simplifying the expression, we get:

y=1+3−4−4y=1+3-4-4

y=−4y=-4

Calculating y for x = 2

To calculate the value of y for x = 2, we will substitute x = 2 into the equation $y=x3+3x2-4x-4$. This gives us:

y=(2)3+3(2)2−4(2)−4y=(2)^3+3(2)^2-4(2)-4

Simplifying the expression, we get:

y=8+12−8−4y=8+12-8-4

y=8y=8

Filling in the Table

Now that we have calculated the value of y for each given x, we can fill in the table accordingly.

x y
-3 -12
-2 8
-1 2
0 -4
1 -4
2 8

Conclusion

In this article, we have explored the process of solving for y in the given equation $y=(x+2)^2(x-1)$. We have used this equation to calculate the value of y for each given x and filled in the table accordingly. We have also simplified the equation to $y=x3+3x2-4x-4$, which represents a cubic function. This equation can have one or three real roots, and we have used it to calculate the value of y for each given x.
Frequently Asked Questions (FAQs) About the Equation y = (x+2)^2(x-1)

In this article, we will answer some frequently asked questions about the equation y = (x+2)^2(x-1). We will cover topics such as the nature of the equation, its graph, and how to solve for y.

Q: What is the nature of the equation y = (x+2)^2(x-1)?

A: The equation y = (x+2)^2(x-1) is a quadratic equation in the form of y = (x+2)^2(x-1). This equation can be expanded to y = (x^2+4x+4)(x-1), which simplifies to y = x3+3x2-4x-4. This equation represents a cubic function, which can have one or three real roots.

Q: What is the graph of the equation y = (x+2)^2(x-1)?

A: The graph of the equation y = (x+2)^2(x-1) is a cubic curve. The graph has a minimum point at x = -2, and it opens upwards. The graph also has a vertical asymptote at x = 1.

Q: How do I solve for y in the equation y = (x+2)^2(x-1)?

A: To solve for y in the equation y = (x+2)^2(x-1), you can substitute the value of x into the equation and simplify the expression. For example, if x = -3, then y = (-3+2)^2(-3-1) = (-1)^2(-4) = 1(-4) = -4.

Q: What is the value of y when x = 0?

A: To find the value of y when x = 0, we can substitute x = 0 into the equation y = (x+2)^2(x-1). This gives us y = (0+2)^2(0-1) = (2)^2(-1) = 4(-1) = -4.

Q: What is the value of y when x = 1?

A: To find the value of y when x = 1, we can substitute x = 1 into the equation y = (x+2)^2(x-1). This gives us y = (1+2)^2(1-1) = (3)^2(0) = 9(0) = 0.

Q: Can I use the equation y = (x+2)^2(x-1) to model real-world situations?

A: Yes, the equation y = (x+2)^2(x-1) can be used to model real-world situations. For example, the equation can be used to model the growth of a population over time, or the cost of a product over time.

Q: How do I graph the equation y = (x+2)^2(x-1)?

A: To graph the equation y = (x+2)^2(x-1), you can use a graphing calculator or a computer program. You can also use a table of values to plot the points on a coordinate plane.

Q: What is the domain of the equation y = (x+2)^2(x-1)?

A: The domain of the equation y = (x+2)^2(x-1) is all real numbers except x = 1. This is because the equation is undefined when x = 1, since it would result in a division by zero.

Q: What is the range of the equation y = (x+2)^2(x-1)?

A: The range of the equation y = (x+2)^2(x-1) is all real numbers except y = 0. This is because the equation is a cubic function, and it can take on any real value except 0.

Conclusion

In this article, we have answered some frequently asked questions about the equation y = (x+2)^2(x-1). We have covered topics such as the nature of the equation, its graph, and how to solve for y. We have also provided examples and explanations to help illustrate the concepts.