Pre-Calculus Honors
9.1: Circles and Parabolas
HW: p.643 (8-24 even, 28, 30, 36, 42)
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Conics
A conic section (or simply conic)
is the intersection of a plane and
a double-napped cone.
Notice in Figure 9.1 that in the
formation of the four basic conics,
the intersecting plane does not
pass through the vertex of the cone.
Basic Conics
Figure 9.1
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Conics
The definition of a circle is the collection of all points (x, y)
that are equidistant from a fixed point (h, k) leads to the
standard equation of a circle
(x – h)2 + (y – k)2 = r2.
Equation of circle
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Example 1 – Finding the Standard Equation of a Circle
The point (1, 4) is on a circle whose center is at (–2, –3), as
shown in Figure 9.4. Write the standard form of the
equation of the circle.
Figure 9.4
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Example 2 – Sketching a Circle
Sketch the circle given by the equation
x2 – 6x + y2 – 2y + 6 = 0
and identify its center and radius.
Solution:
Begin by writing the equation in standard form.
x2 – 6x + y2 – 2y + 6 = 0
(x2 – 6x + __) + (y2 – 2y + __) = –6 + __ + __
Complete the squares
Write in standard form
In this form, you can identify the center and radius of the
circle and then sketch the graph.
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Example 2 – Solution
cont’d
Now graph the function.
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Practice with circles.
cont’d
1. Write the equation of a circle in standard form with
center = (3, 7) and point on the circle = (1, 0).
2. Write the equation of a circle in standard form with
2 7
center = (-3, -1) and diameter =
Identify the center and radius of the circle, then graph.
3. (x – 3)2 + y2 = 8
4. x2 – 14x + y2 + 8y + 40 = 0
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Do Now
Graph 9x2 + 9y2 + 54x – 36y + 17 = 0
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Pre-Calculus Honors
9.1: Circles and Parabolas
HW: p.644 (70 and 76: write in standard form),
(62, 66, 68, 78: graph labeling the vertex and 2
additional points, determine domain and
range).
Copyright © Cengage Learning. All rights reserved.
9
Parabolas
opens up or down b/c x is squared
opens right or left b/c y is squared
Vertex: (h, k); there are other characteristics of a parabola we are not going to find
and sketch in the graph (directrix and focus).
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Parabolas
Rewrite the parabola in standard form:
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Parabolas
Solution: Convert to standard form by completing the square.
–2y = x2 + 2x – 1
1 – 2y = x2 + 2x
__ + 1 – 2y = x2 + 2x + __
2 – 2y = x2 + 2x + 1
–2(y – 1) = (x + 1)2
Create a coefficient of 1 for x2
Isolate the x2 and bx term
Complete the square
Combine like terms
Factor to write in standard form
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Graph the parabola.
Graph the parabola. Label the vertex and 2 additional points
on the parabola. Determine the domain and range.
1. y2 = 3x
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Graph the parabola.
Graph the parabola. Label the vertex and 2 additional points
on the parabola. Determine the domain and range.
2. (x + ½)2 = 4(y – 1)
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Graph the parabola.
Graph the parabola. Label the vertex and 2 additional points
on the parabola. Determine the domain and range.
3. y2 + x + y = 0
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Graph the parabola.
Graph the parabola. Label the vertex and 2 additional points
on the parabola. Determine the domain and range.
4. 3x2 + 6x + y = 4
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