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Chapter 12: Problem 23

Identify the graph of each equation as a parabola, circle, ellipse, or hyperbola, and then sketch the graph. $$ x^{2}=4 y-8 $$

Short Answer

Expert verified
The equation describes a parabola with the vertex at (0, 2), opening upwards.

Step by step solution

01

Identify the Conic Section

Observe the given equation: \[x^{2}=4y-8\]. This equation has the form of a parabola because there is only one squared term. A parabola generally has the format \[x^{2} = 4ay\] or \[y^{2} = 4ax\].
02

Rewrite the Equation in Standard Form

Rewrite the given equation in a more recognizable form of a parabola. To do this, isolate the variable y: \[x^{2} = 4y - 8\] Add 8 to both sides: \[x^{2} + 8 = 4y\] Then divide the entire equation by 4: \[\frac{x^2 + 8}{4} = y\] Simplify to get: \[x^{2} + 8 = 4y \Rightarrow y = \frac{x^{2} + 8}{4} \].
03

Determine the Vertex

The standard form of a vertical parabola is \[x^{2} = 4a(y - k)\] or \[y = \frac{1}{4a} x^2 + k\]. By comparing \[x^2 + 8 = 4y\] (which is equivalent to \[y = \frac{x^2}{4} + 2\]), the vertex (the point \((h,k)\)) is at (0, 2).
04

Sketch the Parabola

Now graph the parabola using the vertex (0, 2) and the fact it opens upwards along the y-axis. Choose a few points on either side of the vertex to see the curvature. For example: When \(x = 2\), \[y = \frac{2^2 + 8}{4}\ = \frac{4 + 8}{4}\ =\frac{12}{4}\ = 3\] Thus, you can plot the point (2, 3).

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Key Concepts

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

parabola
A parabola is one of the four conic sections in geometry. It is formed by the intersection of a right circular conical surface and a plane parallel to a generator of the cone.
Parabolas have interesting properties:
  • All points on a parabola are equidistant from a fixed point called the 'focus' and a fixed line called the 'directrix'.
  • Parabolas have a symmetrical shape, with the line of symmetry passing through the vertex and perpendicular to the directrix.
They often arise in physics and engineering, such as in the paths of projectiles and the design of reflective surfaces in telescopes and satellite dishes.
vertex
The vertex of a parabola is the point where the parabola changes direction. It is the highest or lowest point on the graph, depending on whether the parabola opens upward or downward.
If a parabola opens upward or downward, the vertex is the minimum or maximum point respectively. In mathematical terms:
  • The vertex form for a vertical parabola is \((h,k)\) and is found using the equation = \[y = a(x - h)^2 + k\].
  • For the given equation \[x^2 + 8 = 4y\] the vertex is \((0, 2)\), as deduced by rewriting the equation in standard form.
This tells us that from the vertex, the parabola extends symmetrically in both directions.
graphing equations
Graphing the equation of a parabola involves several steps:
  • First, identify if the parabola opens upward or downward by looking at the coefficient of the squared term.
  • Find the vertex, which is the key point from which the symmetry of the parabola emanates.
  • Choose additional points on either side of the vertex to get a sense of the parabola's shape.
  • Note that for \[y = \frac{x^2 + 8}{4}\], the graph opens upwards since the coefficient of \[x^2\] is positive.

By analyzing upward or downward openings and using symmetry about the vertex, sketching the parabola's graph becomes straightforward.
standard form of parabolas
The standard form of a parabola helps in easily identifying its characteristics such as the vertex, direction of opening, and identifying the conic section.
For vertical parabolas, the standard forms are:
  • \[x^2 = 4a(y-k)\]
  • \[y = a(x-h)^2 + k\].
This form makes it easier to decode essential aspects:
  • Identifies the vertex \((h, k)\)
  • Shows if the parabola opens upwards (positive coefficient) or downwards (negative coefficient).
Applying this to the exercise, the given equation \[x^2 = 4y - 8\] is transformed to \[y = \frac{x^2}{4} + 2\] making it easier to graph and analyze the vertex at \((0, 2)\).

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Most popular questions from this chapter

Suppose that a nonlinear system is composed of equations whose graphs are those described, and the number of points of intersection of the two graphs is as given. Make a sketch satisfying these conditions. (There may be more than one way to do this.) A line and a circle; no points

Solve each system using the elimination method or a combination of the elimination and substitution methods. $$ \begin{array}{l} x^{2}+x y-y^{2}=11 \\ x^{2}-y^{2}=8 \end{array} $$

Solve each system using the substitution method. \(x^{2}-3 x+y^{2}=4\) \(2 x-y=3\)

Graph each system of inequalities. \(5 x-3 y \leq 15\) \(4 x+y \geq 4\)

Graph each system of inequalities. \(x \geq-2\) \(y \leq 4\)
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