Graph The Parabola.$ Y = -2x^2 - 12x - 20 $

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Introduction

Graphing a parabola is an essential skill in mathematics, particularly in algebra and calculus. A parabola is a quadratic function that can be represented in various forms, including the standard form, vertex form, and factored form. In this article, we will focus on graphing the parabola represented by the equation y=−2x2−12x−20y = -2x^2 - 12x - 20. We will explore the properties of the parabola, including its vertex, axis of symmetry, and direction of opening.

Understanding the Equation

The given equation is in the standard form of a quadratic function, which is y=ax2+bx+cy = ax^2 + bx + c. In this case, the coefficients are a=−2a = -2, b=−12b = -12, and c=−20c = -20. The value of aa determines the direction of opening of the parabola, while the values of bb and cc affect the position of the vertex.

Determining the Vertex

The vertex of a parabola is the point where the parabola changes direction. To find the vertex, we can use the formula x=−b2ax = -\frac{b}{2a}. Plugging in the values of aa and bb, we get:

x=−−122(−2)x = -\frac{-12}{2(-2)} x=−−12−4x = -\frac{-12}{-4} x=−3x = -3

Now that we have the x-coordinate of the vertex, we can find the y-coordinate by plugging the value of xx into the original equation:

y=−2(−3)2−12(−3)−20y = -2(-3)^2 - 12(-3) - 20 y=−2(9)+36−20y = -2(9) + 36 - 20 y=−18+36−20y = -18 + 36 - 20 y=−2y = -2

Therefore, the vertex of the parabola is at the point (−3,−2)(-3, -2).

Axis of Symmetry

The axis of symmetry is a vertical line that passes through the vertex of the parabola. Since the vertex is at (−3,−2)(-3, -2), the axis of symmetry is the line x=−3x = -3.

Direction of Opening

The direction of opening of the parabola is determined by the value of aa. Since a=−2a = -2, the parabola opens downward.

Graphing the Parabola

To graph the parabola, we can start by plotting the vertex at (−3,−2)(-3, -2). Then, we can use the axis of symmetry to draw a vertical line through the vertex. Next, we can plot a few points on either side of the axis of symmetry to get an idea of the shape of the parabola.

Plotting Points

To plot points on the parabola, we can use the equation y=−2x2−12x−20y = -2x^2 - 12x - 20. We can choose values of xx and plug them into the equation to find the corresponding values of yy. For example, let's choose x=−4x = -4 and x=−2x = -2.

For x=−4x = -4:

y=−2(−4)2−12(−4)−20y = -2(-4)^2 - 12(-4) - 20 y=−2(16)+48−20y = -2(16) + 48 - 20 y=−32+48−20y = -32 + 48 - 20 y=−4y = -4

For x=−2x = -2:

y=−2(−2)2−12(−2)−20y = -2(-2)^2 - 12(-2) - 20 y=−2(4)+24−20y = -2(4) + 24 - 20 y=−8+24−20y = -8 + 24 - 20 y=−4y = -4

Therefore, the points (−4,−4)(-4, -4) and (−2,−4)(-2, -4) lie on the parabola.

Graphing the Parabola

Using the points we plotted earlier, we can draw a smooth curve through the points to get the graph of the parabola.

Conclusion

Q&A: Graphing the Parabola

Q: What is the vertex of the parabola represented by the equation y=−2x2−12x−20y = -2x^2 - 12x - 20? A: The vertex of the parabola is at the point (−3,−2)(-3, -2).

Q: What is the axis of symmetry of the parabola? A: The axis of symmetry is the line x=−3x = -3.

Q: In which direction does the parabola open? A: The parabola opens downward.

Q: How do I graph the parabola? A: To graph the parabola, start by plotting the vertex at (−3,−2)(-3, -2). Then, use the axis of symmetry to draw a vertical line through the vertex. Next, plot a few points on either side of the axis of symmetry to get an idea of the shape of the parabola.

Q: How do I plot points on the parabola? A: To plot points on the parabola, choose values of xx and plug them into the equation y=−2x2−12x−20y = -2x^2 - 12x - 20 to find the corresponding values of yy.

Q: What are some common mistakes to avoid when graphing a parabola? A: Some common mistakes to avoid when graphing a parabola include:

  • Not plotting the vertex correctly
  • Not using the axis of symmetry correctly
  • Not plotting enough points to get an accurate idea of the shape of the parabola
  • Not using a ruler or other straightedge to draw the graph

Q: How can I check my work when graphing a parabola? A: To check your work when graphing a parabola, you can:

  • Use a calculator to graph the parabola and compare it to your hand-drawn graph
  • Check that the vertex and axis of symmetry are correct
  • Check that the parabola opens in the correct direction
  • Check that the points you plotted lie on the parabola

Q: What are some real-world applications of graphing a parabola? A: Graphing a parabola has many real-world applications, including:

  • Modeling the trajectory of a projectile
  • Modeling the motion of an object under the influence of gravity
  • Modeling the growth of a population
  • Modeling the spread of a disease

Q: How can I use graphing a parabola to solve problems? A: Graphing a parabola can be used to solve problems in a variety of ways, including:

  • Finding the maximum or minimum value of a function
  • Finding the intersection of two or more functions
  • Finding the area under a curve
  • Finding the volume of a solid

Conclusion

Graphing a parabola is an essential skill in mathematics. By understanding the properties of the parabola, including its vertex, axis of symmetry, and direction of opening, we can graph the parabola accurately. In this article, we graphed the parabola represented by the equation y=−2x2−12x−20y = -2x^2 - 12x - 20. We answered common questions about graphing a parabola and provided tips and tricks for graphing a parabola accurately.