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Chapter 11: Problem 14

Give equations of parabolas. Find each parabola's focus and directrix. Then sketch the parabola. Include the focus and directrix in your sketch. \(y=-8 x^{2}\)

Short Answer

Expert verified
Focus: \((0, -\frac{1}{32})\); Directrix: \(y = \frac{1}{32}\); Opens downward.

Step by step solution

01

Identify the equation form

The given parabola equation is in the form of \(y = ax^2 + bx + c\). For \(y = -8x^2\), we notice it has no linear (\(x\)) or constant term. It is a vertical parabola with the vertex at the origin \((0, 0)\).
02

Determine the direction of the parabola

Since \(a = -8\) is negative, the parabola opens downward. This means the vertex will be the highest point on the graph.
03

Find the focus of the parabola

The formula for the focus of a vertical parabola \(y = ax^2\) is \((0, \frac{1}{4a})\). Here, \(a = -8\), so the focus is at \(\left(0, \frac{1}{4(-8)}\right) = (0, -\frac{1}{32})\).
04

Find the directrix of the parabola

The directrix of a vertical parabola \(y = ax^2\) is given by the equation \(y = -\frac{1}{4a}\). Substituting \(a = -8\), the directrix is \(y = \frac{1}{32}\).
05

Sketch the parabola including the focus and directrix

On a coordinate plane, draw the vertex at the origin (0, 0). Since the parabola opens downward, illustrate it as a U-shaped curve concave down. Mark the focus at \((0, -\frac{1}{32})\) and draw a horizontal line for the directrix at \(y = \frac{1}{32}\). Ensure the parabola is symmetrical with respect to the y-axis.

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Key Concepts

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

Focus of a Parabola
In the study of parabolas, the focus is a critical point that helps determine the shape and orientation of the curve. It is one of the fixed points used in the parabola's definition.
The focus of a vertical parabola, like the one described by the equation \(y = -8x^2\), lies along the axis of symmetry on the y-axis. It is not at the vertex but slightly away from it. The precise location of the focus is determined by the parameter \(a\) in the equation.
  • The formula to find the focus for a vertical parabola is \((0, \frac{1}{4a})\).
  • In the current example, with \(a = -8\), the focus is located at \((0, -\frac{1}{32})\).
This point plays a crucial role as it guides the curve to open outward, ensuring all points on the parabola are equidistant from the focus and the directrix.
Directrix of a Parabola
The directrix of a parabola is an essential geometric component that works together with the focus to define the parabola's shape. It is a fixed line opposite the focus, ensuring the parabola's defining property: that any point on the curve is equidistant from the focus and the directrix.
For a vertical parabola such as \(y = -8x^2\), the directrix is parallel to the x-axis and lies symmetrically in relation to the focus on the y-axis.
  • The directrix can be calculated using the formula: \(y = -\frac{1}{4a}\).
  • With an \(a\) value of \(-8\), the directrix is found at \(y = \frac{1}{32}\).
This straight line helps to create the balance needed for the parabola鈥檚 shape, ensuring the curve opens downward due to its negative orientation.
Vertex of a Parabola
Every parabola has a vertex, the point where the curve changes direction. In simple terms, when dealing with vertical parabolas like \(y = -8x^2\), this is the peak if the parabola opens downwards or the lowest point if it opens upwards.
For the equation \(y = -8x^2\), the vertex is found at the origin, \((0, 0)\). It is important as it serves as the symmetric middle of the parabola.
  • In vertical parabolas without a linear term, such as in this case, the vertex is always at \((0, 0)\).
  • The vertex gives you a central point from which the parabola stretches out symmetrically along the y-axis.
Understanding the vertex helps in sketching the curve accurately and helps highlight the symmetry inherent in parabolas.
Vertical Parabolas
Vertical parabolas are a special kind of parabola that opens either upwards or downwards. Their standard form is \(y = ax^2 + bx + c\). When the equation is simplified, such as \(y = -8x^2\), we see clearly how they behave.
Key characteristics include:
  • Orientation: Determined by the sign of \(a\). If \(a < 0\), the parabola opens downwards; if \(a > 0\), it opens upwards.
  • Vertex: Found at \((0, 0)\) when there are no additional terms.
  • Symmetry: Vertical parabolas are symmetrically balanced with respect to the y-axis.
Given that \(a = -8\) in our example, this parabola is concave down, forming a U-shaped curve with its vertex as the peak. These properties are crucial for graphing and understanding the parabola's trajectory.

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Most popular questions from this chapter

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