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Chapter 10: Problem 36

Identify whether equation, when graphed, will be a parabola, circle, ellipse, or hyperbola. Sketch the graph of equation. If a parabola, label the vertex. If a circle, label the center and note the radius. If an ellipse, label the center. If a hyperbola, label the \(x\) - or \(y\) -intercepts. \(y=x^{2}+4\)

Short Answer

Expert verified
The equation is a parabola with vertex at \((0, 4)\).

Step by step solution

01

Recognize the Type of Equation

The given equation is in the form of \( y = x^2 + 4 \). This is a quadratic equation in two variables, which indicates it is a parabola. Parabolas have an equation of the form \( y = ax^2 + bx + c \). Since there is a single squared term and no terms involving the product of \( x \) and \( y \), the graph is a parabola.
02

Identify the Parabola's Orientation

The equation \( y = x^2 + 4 \) is in the standard form \( y = x^2 + c \). This indicates that the parabola opens upwards, as the coefficient of \( x^2 \) is positive.
03

Determine the Vertex of the Parabola

The vertex of a standard parabola \( y = ax^2 + bx + c \) can be found using the formula \( x = -\frac{b}{2a} \). In our equation, \( b = 0 \), so \( x = -\frac{0}{2 \cdot 1} = 0 \). Substituting \( x = 0 \) back into the equation gives \( y = (0)^2 + 4 = 4 \). Thus, the vertex is at \( (0, 4) \).
04

Sketch the Graph

Draw the Cartesian plane. Plot the vertex at \( (0, 4) \). Since the parabola opens upwards, draw a symmetric curve extending from the vertex downward and outward along the y-axis, showing the general shape of a parabola.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Quadratic Equations
Quadratic equations are essential foundations in mathematics, specifically when dealing with parabolas. These equations take the form \( y = ax^2 + bx + c \), where \( a \), \( b \), and \( c \) are constants, and \( a eq 0 \). The key feature of a quadratic equation is the squared term, which introduces the concept of a parabolic curve.
Quadratic equations can be solved using various methods, such as factoring, completing the square, or applying the quadratic formula. The solutions to these equations, known as roots or zeros, represent the points where the parabola intersects the x-axis.
  • **Vertex form**: A quadratic equation can also be rewritten in vertex form, \( y = a(x-h)^2 + k \), where \( (h, k) \) is the vertex of the parabola.
  • **Importance**: Understanding quadratic equations is crucial for graphing and manipulating parabolas in various mathematical applications.
Vertex of a Parabola
The vertex of a parabola is a critical point where the curve changes direction. It is the highest or lowest point of the parabola, depending on its orientation. For the equation \( y = ax^2 + bx + c \), the vertex can be found using the formula \( x = -\frac{b}{2a} \).
The vertex offers significant information about the parabola, such as its axis of symmetry, which is a vertical line running through the vertex.
  • **Coordinates of the vertex**: Once the \( x \) value is calculated using the formula, substitute it back into the equation to find the corresponding \( y \) value. These coordinates together define the vertex \( (x, y) \).
  • **Importance of the vertex**: The vertex not only aids in graphing the parabola but also provides insights into its maximum or minimum values.
Graphing Parabolas
Graphing parabolas requires understanding their shape and key features, like the vertex and axis of symmetry. Parabolas can either open upwards or downwards based on the sign of the \( a \) coefficient in the equation \( y = ax^2 + bx + c \). If \( a > 0 \), the parabola opens upward, and if \( a < 0 \), it opens downward.
To graph a parabola, follow these steps:
  • **Plot the vertex**: Start by locating the vertex on the coordinate plane, as this is a pivotal point for the curve.
  • **Draw the axis of symmetry**: This vertical line passes through the vertex, guiding the symmetry of the parabola.
  • **Plot additional points**: Choose values for \( x \) around the vertex and calculate corresponding \( y \) values to find more points on the graph.
  • **Sketch the curve**: Use the vertex and the additional points to draw a smooth, symmetric curve through these points, ensuring it follows the appropriate direction based on the sign of \( a \).
Conic Sections
Conic sections are curves obtained by intersecting a cone with a plane. This intersection results in various shapes like parabolas, circles, ellipses, and hyperbolas. Each conic section has distinctive characteristics and equations. For example, a parabola, like \( y = x^2 + 4 \), is one such conic section characterized by an equation with a single squared variable term.
Conic sections are important in many aspects of geometry and algebra. Understanding them can help solve problems related to their properties and applications.
  • **Different types**: Conic sections include parabolas, circles, ellipses, and hyperbolas, each with its unique algebraic expressions and graphical representations.
  • **Relevance**: These sections have practical applications in fields such as physics, engineering, and astronomy, where they can model natural phenomena and man-made structures.

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Most popular questions from this chapter

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The orbits of stars, planets, comets, asteroids, and satellites all have the shape of one of the conic sections. Astronomers use a measure called eccentricity to describe the shape and elongation of an orbital path. For the circle and ellipse, eccentricity e is calculated with the formula \(e=\frac{c}{d},\) where \(c^{2}=\left|a^{2}-b^{2}\right|\) and \(d\) is the larger value of a or b. For a hyperbola, eccentricity e is calculated with the formula \(e=\frac{c}{d},\) where \(c^{2}=a^{2}+b^{2}\) and the value of \(d\) is equal to a if the hyperbola has \(x\) -intercepts or equal to b if the hyperbola has \(y\) -intercepts. A. \(\frac{x^{2}}{36}-\frac{y^{2}}{13}=1\) B. \(\frac{x^{2}}{4}+\frac{y^{2}}{4}=1\) C. \(\frac{x^{2}}{25}+\frac{y^{2}}{16}=1\) D. \(\frac{y^{2}}{25}-\frac{x^{2}}{39}=1\) G. \(\frac{x^{2}}{16}-\frac{y^{2}}{65}=1\) E. \(\frac{x^{2}}{17}+\frac{y^{2}}{81}=1\) F. \(\frac{x^{2}}{36}+\frac{y^{2}}{36}=1\) H. \(\frac{x^{2}}{144}+\frac{y^{2}}{140}=1\) What do you notice about the values of \(e\) for the equations you identified as circles?

Comets orbit the sun in elongated ellipses. Consider the sun as the origin of a rectangular coordinate system. Suppose that the equation of the path of the comet is $$\frac{(x-1,782,000,000)^{2}}{3.42 \times 10^{23}}+\frac{(y-356,400,000)^{2}}{1.368 \times 10^{22}}=1$$ Find the center of the path of the comet.
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