Graph The Parabola Y = − X 2 − 2 X + 1 Y = -x^2 - 2x + 1 Y = − X 2 − 2 X + 1 .Plot Five Points On The Parabola: - The Vertex,- Two Points To The Left Of The Vertex,- And Two Points To The Right Of The Vertex.

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Introduction

In mathematics, a parabola is a quadratic curve that can be represented by the equation $y = ax^2 + bx + c$. The graph of a parabola is a U-shaped curve that opens upwards or downwards, depending on the value of $a$. In this article, we will graph the parabola $y = -x^2 - 2x + 1$ and plot five points on the parabola, including the vertex, two points to the left of the vertex, and two points to the right of the vertex.

Finding the Vertex

The vertex of a parabola is the point on the curve where it changes direction. To find the vertex of the parabola $y = -x^2 - 2x + 1$, we can use the formula $x = -\frac{b}{2a}$. In this case, $a = -1$ and $b = -2$, so we have:

$x = -\frac{-2}{2(-1)} = -\frac{-2}{-2} = -1$

Now that we have the x-coordinate of the vertex, we can find the y-coordinate by plugging $x = -1$ into the equation of the parabola:

$y = -(-1)^2 - 2(-1) + 1 = -1 + 2 + 1 = 2$

Therefore, the vertex of the parabola $y = -x^2 - 2x + 1$ is the point $(-1, 2)$.

Plotting Points on the Parabola

To plot points on the parabola, we need to choose values of $x$ and plug them into the equation of the parabola to find the corresponding values of $y$. Let's choose $x = -2$, $x = -1$, $x = 0$, $x = 1$, and $x = 2$.

For $x = -2$, we have:

$y = -(-2)^2 - 2(-2) + 1 = -4 + 4 + 1 = 1$

For $x = -1$, we have:

$y = -(-1)^2 - 2(-1) + 1 = -1 + 2 + 1 = 2$

For $x = 0$, we have:

$y = -(0)^2 - 2(0) + 1 = 1$

For $x = 1$, we have:

$y = -(1)^2 - 2(1) + 1 = -1 - 2 + 1 = -2$

For $x = 2$, we have:

$y = -(2)^2 - 2(2) + 1 = -4 - 4 + 1 = -7$

Therefore, the five points on the parabola are $(-2, 1)$, $(-1, 2)$, $(0, 1)$, $(1, -2)$, and $(2, -7)$.

Graphing the Parabola

To graph the parabola, we can use a graphing calculator or a computer program to plot the points we have found. The graph of the parabola $y = -x^2 - 2x + 1$ is a U-shaped curve that opens downwards.

Conclusion

In this article, we have graphed the parabola $y = -x^2 - 2x + 1$ and plotted five points on the parabola, including the vertex, two points to the left of the vertex, and two points to the right of the vertex. We have also found the vertex of the parabola using the formula $x = -\frac{b}{2a}$. The graph of the parabola is a U-shaped curve that opens downwards.

Mathematical Representation of the Parabola

The parabola $y = -x^2 - 2x + 1$ can be represented mathematically as a quadratic equation in the form $y = ax^2 + bx + c$. The coefficients of the quadratic equation are $a = -1$, $b = -2$, and $c = 1$.

Graphing the Parabola using a Graphing Calculator

To graph the parabola using a graphing calculator, we can enter the equation of the parabola and use the calculator's graphing function to plot the points. The graph of the parabola will be a U-shaped curve that opens downwards.

Graphing the Parabola using a Computer Program

To graph the parabola using a computer program, we can use a programming language such as Python or MATLAB to plot the points. The graph of the parabola will be a U-shaped curve that opens downwards.

Properties of the Parabola

The parabola $y = -x^2 - 2x + 1$ has several properties that can be used to analyze its behavior. Some of these properties include:

• Vertex: The vertex of the parabola is the point $(-1, 2)$.
• Axis of Symmetry: The axis of symmetry of the parabola is the vertical line $x = -1$.
• Direction: The parabola opens downwards.
• Intercepts: The parabola has no x-intercepts, but it has a y-intercept at the point $(0, 1)$.

Real-World Applications of the Parabola

The parabola $y = -x^2 - 2x + 1$ has several real-world applications, including:

• Projectile Motion: The parabola can be used to model the trajectory of a projectile under the influence of gravity.
• Optics: The parabola can be used to design optical systems, such as telescopes and microscopes.
• Engineering: The parabola can be used to design structures, such as bridges and buildings.

Conclusion

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Frequently Asked Questions

Q: What is the vertex of the parabola $y = -x^2 - 2x + 1$?

A: The vertex of the parabola $y = -x^2 - 2x + 1$ is the point $(-1, 2)$.

Q: How do I find the vertex of a parabola?

A: To find the vertex of a parabola, you can use the formula $x = -\frac{b}{2a}$. In this case, $a = -1$ and $b = -2$, so we have:

$x = -\frac{-2}{2(-1)} = -\frac{-2}{-2} = -1$

Now that we have the x-coordinate of the vertex, we can find the y-coordinate by plugging $x = -1$ into the equation of the parabola:

$y = -(-1)^2 - 2(-1) + 1 = -1 + 2 + 1 = 2$

Therefore, the vertex of the parabola $y = -x^2 - 2x + 1$ is the point $(-1, 2)$.

Q: How do I plot points on the parabola?

A: To plot points on the parabola, you need to choose values of $x$ and plug them into the equation of the parabola to find the corresponding values of $y$. Let's choose $x = -2$, $x = -1$, $x = 0$, $x = 1$, and $x = 2$.

For $x = -2$, we have:

$y = -(-2)^2 - 2(-2) + 1 = -4 + 4 + 1 = 1$

For $x = -1$, we have:

$y = -(-1)^2 - 2(-1) + 1 = -1 + 2 + 1 = 2$

For $x = 0$, we have:

$y = -(0)^2 - 2(0) + 1 = 1$

For $x = 1$, we have:

$y = -(1)^2 - 2(1) + 1 = -1 - 2 + 1 = -2$

For $x = 2$, we have:

$y = -(2)^2 - 2(2) + 1 = -4 - 4 + 1 = -7$

Therefore, the five points on the parabola are $(-2, 1)$, $(-1, 2)$, $(0, 1)$, $(1, -2)$, and $(2, -7)$.

Q: How do I graph the parabola using a graphing calculator?

A: To graph the parabola using a graphing calculator, you can enter the equation of the parabola and use the calculator's graphing function to plot the points. The graph of the parabola will be a U-shaped curve that opens downwards.

Q: How do I graph the parabola using a computer program?

A: To graph the parabola using a computer program, you can use a programming language such as Python or MATLAB to plot the points. The graph of the parabola will be a U-shaped curve that opens downwards.

Q: What are some real-world applications of the parabola?

A: The parabola $y = -x^2 - 2x + 1$ has several real-world applications, including:

• Projectile Motion: The parabola can be used to model the trajectory of a projectile under the influence of gravity.
• Optics: The parabola can be used to design optical systems, such as telescopes and microscopes.
• Engineering: The parabola can be used to design structures, such as bridges and buildings.

Q: What are some properties of the parabola?

A: The parabola $y = -x^2 - 2x + 1$ has several properties, including:

• Vertex: The vertex of the parabola is the point $(-1, 2)$.
• Axis of Symmetry: The axis of symmetry of the parabola is the vertical line $x = -1$.
• Direction: The parabola opens downwards.
• Intercepts: The parabola has no x-intercepts, but it has a y-intercept at the point $(0, 1)$.

Q: How do I find the equation of a parabola?

A: To find the equation of a parabola, you can use the formula $y = ax^2 + bx + c$. In this case, $a = -1$, $b = -2$, and $c = 1$, so we have:

$y = -x^2 - 2x + 1$

Therefore, the equation of the parabola is $y = -x^2 - 2x + 1$.

Q: How do I graph a parabola?

A: To graph a parabola, you can use a graphing calculator or a computer program to plot the points. The graph of the parabola will be a U-shaped curve that opens upwards or downwards, depending on the value of $a$.

Q: What are some common mistakes when graphing a parabola?

A: Some common mistakes when graphing a parabola include:

• Incorrectly identifying the vertex: Make sure to use the formula $x = -\frac{b}{2a}$ to find the x-coordinate of the vertex.
• Incorrectly plotting points: Make sure to plug the values of $x$ into the equation of the parabola to find the corresponding values of $y$.
• Incorrectly graphing the parabola: Make sure to use a graphing calculator or a computer program to plot the points, and make sure the graph is a U-shaped curve that opens upwards or downwards.

Q: How do I check my work when graphing a parabola?

A: To check your work when graphing a parabola, you can:

• Verify the vertex: Make sure the vertex is at the correct point.
• Verify the axis of symmetry: Make sure the axis of symmetry is at the correct line.
• Verify the direction: Make sure the parabola opens upwards or downwards, depending on the value of $a$.
• Verify the intercepts: Make sure the parabola has the correct x-intercepts and y-intercept.

By following these steps and checking your work, you can ensure that your graph of the parabola is accurate and complete.