Explanations 
Textbooks 
Flashcards 
91影视 AI 
Notes 
Study Plans 
Study Sets 
Find a degree 
Magazine Chapter 21: Problem 25
Identify the curve represented by each of the given equations. Determine the appropriate important quantities for the curve and sketch the graph. $$x^{2}=8(y-x-2)$$
Short Answer
Expert verified
The curve is a vertical parabola with vertex \((-4, 0)\), opening upward.
Step by step solution
01
Identify the Type of Conic Section
The given equation is \(x^2 = 8(y - x - 2)\). We recognize that it involves only the square of \(x\), suggesting it might describe a parabola. Parabolas have equations of the form \((x-h)^2 = 4p(y-k)\) or \((y-k)^2 = 4p(x-h)\).
02
Simplify and Rearrange the Equation
First, expand and arrange the equation in a recognizable form. Begin by distributing the right-hand side: \(x^2 = 8y - 8x - 16\). Rearrange to gather like terms: \(x^2 + 8x = 8y - 16\).
03
Complete the Square
To convert \(x^2 + 8x\) into a perfect square, notice that completing the square involves \((x^2 + 8x + 16 - 16)\), resulting in \((x+4)^2 - 16\). Hence, the equation becomes \((x+4)^2 - 16 = 8y - 16\).
04
Further Simplify the Equation
Simplify \((x+4)^2 - 16 = 8y - 16\) by moving the constant terms to the right side: \((x+4)^2 = 8y\). This fits the form of a vertical parabola equation \((x - h)^2 = 4p(y - k)\), where \((h, k) = (-4, 0)\) and \(4p = 8\) implies \(p = 2\).
05
Interpret the Important Quantities
For the vertex form \((x+4)^2 = 8(y - 0)\), the vertex is \((-4, 0)\). Since \(p = 2\), the parabola opens upward with the focus at \((-4, 2)\) and the directrix is given by \(y = -2\).
06
Sketch the Graph
On a coordinate plane, plot the vertex of the parabola at \((-4, 0)\). The axis of symmetry is vertical. Plot the focus at \((-4, 2)\) and draw the directrix \(y = -2\). Draw the parabola passing through \((-4, 0)\), opening upward.
Unlock Step-by-Step Solutions & Ace Your Exams!
-
Full Textbook Solutions
Get detailed explanations and key concepts
-
Unlimited Al creation
Al flashcards, explanations, exams and more...
-
Ads-free access
To over 500 millions flashcards
-
Money-back guarantee
We refund you if you fail your exam.
Over 30 million students worldwide already upgrade their learning with 91影视!
Key Concepts
These are the key concepts you need to understand to accurately answer the question.
Parabola
A parabola is a unique curve that is part of the family of conic sections, which also includes ellipses, circles, and hyperbolas. Parabolas have a distinctive U-shape and are defined mathematically by quadratic equations. They have a single axis of symmetry and contain a vertex, which is the point where the parabola turns.
- In their most basic form, a parabola's equation can be written as \(y = ax^2 + bx + c\), especially when the parabola opens upwards or downwards. In cases where it opens sideways, the general form might look like \(x = ay^2 + by + c\).
- Every parabola has important features including the vertex, focus, directrix, and axis of symmetry, which all play a role in describing its specific orientation and shape.
Completing the Square
Completing the square is a crucial algebraic technique often used to transform a quadratic equation into a more manageable form. The goal is to create a perfect square trinomial, which can be factored efficiently.
To complete the square for a quadratic term such as \(x^2 + bx\), you can use the following process:
To complete the square for a quadratic term such as \(x^2 + bx\), you can use the following process:
- Start by finding half of the coefficient of the linear term (\(b\)), then square it. For example, if the equation is \(x^2 + 8x\), take half of 8 to get 4, and square it to get 16.
- Add and subtract this number inside the equation to keep it balanced. Thus, \(x^2 + 8x\) becomes \(x^2 + 8x + 16 - 16\).
- Factor the perfect square trinomial, resulting in \((x + 4)^2\) in this instance.
Vertex Form
The vertex form of a quadratic equation is a particularly useful version of the parabola equation that highlights the vertex's coordinates and allows for easy graphing.
The vertex form is written as:
The vertex form is written as:
- \((x-h)^2 = 4p(y-k)\) for parabolas that open upwards or downwards, and \((y-k)^2 = 4p(x-h)\) for those that open sideways.
- Here, \(h, k\) are the coordinates of the vertex, and the value \(p\) determines how "stretched" or "compressed" the parabola is.
Graphing Parabolas
Graphing a parabola effectively involves understanding and plotting several critical components related to its standard form.
These critical components include:
These critical components include:
- The vertex, which serves as the "peak" or "lowest" point of the parabola depending on its orientation.
- The axis of symmetry, a vertical or horizontal line passing through the vertex, dividing the parabola into two mirror-image halves.
- The direction in which it opens, which is determined by the coefficient of the squared term and the value of \(p\) in the vertex form.
- The focus and directrix, which are secondary features that help in clearly defining the parabola鈥檚 shape and ensuring the parabola correctly shows the locus of points equidistant from these features.
