Skip to main content
Ch. 7 - Conic Sections
Blitzer - College Algebra 8th Edition
Blitzer8th EditionCollege AlgebraISBN: 9780136970514Not the one you use?Change textbook
All textbooksBlitzer 8th EditionCh. 7 - Conic SectionsProblem 43
Chapter 8, Problem 43

Convert each equation to standard form by completing the square on x or y. Then find the vertex, focus, and directrix of the parabola. Finally, graph the parabola. x2 - 2x - 4y + 9 =0

Verified step by step guidance
1
Start with the given equation: \(x^2 - 2x - 4y + 9 = 0\).
Group the \(x\) terms together and move the \(y\) term and constant to the other side: \(x^2 - 2x = 4y - 9\).
Complete the square for the \(x\) terms. Take half of the coefficient of \(x\) (which is \(-2\)), square it, and add it to both sides: half of \(-2\) is \(-1\), and \((-1)^2 = 1\). So, add \(1\) to both sides: \(x^2 - 2x + 1 = 4y - 9 + 1\).
Rewrite the left side as a perfect square: \((x - 1)^2 = 4y - 8\). Then isolate \(y\): \((x - 1)^2 = 4(y - 2)\).
Identify the vertex from the standard form \((x - h)^2 = 4p(y - k)\), where the vertex is at \((h, k)\). Here, \(h = 1\) and \(k = 2\). Use the value of \$4p\( to find \)p\(, which helps determine the focus and directrix. The focus is at \)(h, k + p)\( and the directrix is the line \)y = k - p$.

Verified video answer for a similar problem:

This video solution was recommended by our tutors as helpful for the problem above.
Video duration:
4m
Was this helpful?

Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

Completing the Square

Completing the square is a method used to rewrite quadratic expressions in the form (x - h)^2 = k, which helps convert equations into standard form. This technique involves adding and subtracting a constant to create a perfect square trinomial, making it easier to identify key features of conic sections like parabolas.
Recommended video:
06:24
Solving Quadratic Equations by Completing the Square

Standard Form of a Parabola

The standard form of a parabola with a vertical axis is (x - h)^2 = 4p(y - k), where (h, k) is the vertex and p determines the distance to the focus and directrix. Converting to this form allows for straightforward identification of the vertex, focus, and directrix, which are essential for graphing the parabola.
Recommended video:
5:33
Parabolas as Conic Sections

Vertex, Focus, and Directrix of a Parabola

The vertex is the parabola's turning point, the focus is a fixed point inside the curve, and the directrix is a line outside the curve. The parabola is the set of points equidistant from the focus and directrix. Knowing these elements helps in accurately graphing and understanding the parabola's shape and orientation.
Recommended video:
7:18
Horizontal Parabolas Example 1
Related Practice
Textbook Question

Graph each ellipse and give the location of its foci. (x +3)²/9 + (y -2)² = 1

Textbook Question

Find the standard form of the equation of the parabola satisfying the given conditions. Focus: (12,0); Directrix: x=-12

1
views
Textbook Question

Convert each equation to standard form by completing the square on x and y. Then graph the hyperbola. Locate the foci and find the equations of the asymptotes. x2−y2−2x−4y−4=0x^2−y^2−2x−4y−4=0

Textbook Question

Find the standard form of the equation of the parabola satisfying the given conditions. Focus: (0,-11); Directrix: y=11

Textbook Question

Graph each ellipse and give the location of its foci. (x − 4)²/9 + (y +2)² /25= 1

2
views
Textbook Question

Graph each ellipse and give the location of its foci. x²/25 + (y -2)² /36= 1