Secondary 2
Chapter 18
Secondary II
Chapter 18 – Circles and Parabolas
Date
Section
2014/2015
Assignment
Concept
- Worksheet 18.1
Circles and Polygons on the
Coordinate Plane
A: 4/10
B: 4/13
18.1
A: 4/14
B: 4/15
18.2 & 18.3
- Worksheet 18.2 & 18.3
Deriving the Equation for a Circle
Determining Points on a Circle
A: 4/16
B: 4/17
18.4 & 18.5
- Worksheet 18.4 & 18.5
Equation of a Parabola
More with Parabolas
A: 4/20
B: 4/21
Review
A: 4/22
B: 4/23
TEST
Late and absent work will be due on the day of the review (absences must be excused). The review assignment
must be turned in on test day. All required work must be complete to get the curve on the test.
Remember, you are still required to take the test on the scheduled day even if you miss the review, so come
prepared. If you are absent on test day, you will be required to take the test in class the day you return. You will
not receive the curve on the test if you are absent on test day unless you take the test prior to your absence.
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Secondary 2
Chapter 18
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Secondary 2
Chapter 18
Chapter 18: Circles and Parabolas
18.1 – The Coordinate Plane
(Standard: G.GPE.4)
This lesson provides you an opportunity to review several familiar theorems while classifying
polygons formed on a coordinate plane. Circles and polygons located on a coordinate plane enable
you to calculate distances, slopes, and equations of lines.
1. Lauren and Jamie were practicing their discus
throws. They each spun around in a circle to gain
speed and then released their discus at a tangent to
the circular spin. Jamie is left-handed, so she spun
clockwise and released her discus at point T shown.
Lauren is right-handed. She spun counterclockwise
and released at point N. Both of the girls’ throws
landed at point A.
a. Use the given information to algebraically show
that if two tangents are drawn from the same
point on the exterior of a circle, then the tangent
segments are congruent.
b. What conclusions can you make about Lauren’s and Jamie’s discus throws?
Reminder:
Square:
-
All sides are ________________________.
-
Opposite sides are ____________________ which means slopes are ___________________.
-
Each angle is _____________________ which means slopes are _______________________.
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Chapter 18
Rectangle:
-
Opposite sides are ____________________ and ______________________.
-
Each angle is ___________________ which means that slopes are __________________.
Parallelogram:
-
Opposite sides are ______________________ and _________________________.
Rhombus:
-
All sides are _____________________.
-
Opposite sides are_____________________________.
Parallel lines have the ___________________ slopes.
Perpendicular lines have ______________________, _______________________ slopes.
Slope Formula:
Distance Formula:
Midpoint Formula:
Steps to Classifying Shapes on a Coordinate Plane
1. Find the midpoints of each line,
2. Find the slopes of each line,
3. Find the distance of each line,
4. Classify the shape using the information from steps 1-3.
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Secondary 2
Chapter 18
2. Four points on the circle were connected to form a square.
Classify the polygon formed by connecting the midpoints of
the sides of the square.
3. Choose four points on the circle to form a quadrilateral that is not a square. Provide the
coordinates and labels for each. Classify the polygon formed by connecting the midpoints of the
sides of the quadrilateral.
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Secondary 2
Chapter 18
4. Determine the shape formed by connecting the
midpoints of the sides of an isosceles trapezoid.
a. Draw an isosceles trapezoid. Choose
reasonable coordinates for each vertex.
b. Label the vertices of the trapezoid.
c. Determine the coordinates of the midpoint
of each side.
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Secondary 2
Chapter 18
Additional Notes
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Secondary 2
Chapter 18
18.2 & 18.3 – Deriving the Equation for a Circle & Determining Points on a Circle
(Standard: G.GPE.1, G.SRT.8, G.GPE.4)
Recall that a circle is the set of points on
a plane equidistant from a given point.
If this circle is drawn on a coordinate plane, with its center point located at the origin and a point
(x, y) on the circle, it is possible to write an algebraic equation for the circle.
1. Label the sides of the right triangle formed. Then, use the
Pythagorean Theorem to solve for 𝑟 2 .
2. Next, consider a circle with its center located at point
(h, k) and a point (x, y) on the circle. Label the sides of
the right triangle formed. Then, use the Pythagorean
Theorem to solve for 𝑟 2 .
The standard form of the equation of a circle centered at (h, k) with radius r can be expressed as
(𝑥 − ℎ)2 + (𝑦 − 𝑘)2 = 𝑟 2
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Secondary 2
Chapter 18
3. Write an equation for:
a. a circle with center
at the origin and r = 8.
b. circle Q.
4. Circle R is represented by the equation (𝑥 − 4)2 + (𝑦 + 1)2 = 36.
a. Determine the equation of a circle that has the same center as circle R but whose
circumference is twice that of circle R.
b. Determine the equation of a circle that has the same center as circle R but whose
circumference is three times that of circle R.
There are an infinite number of points located on circle A.
To determine the coordinates of other points located on
circle A, you can use the Pythagorean Theorem.
5. Use the Pythagorean Theorem to determine if point B
(4, 3) lies on circle A, and then explain
your reasoning.
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Secondary 2
Chapter 18
6. Consider circle D centered at the origin with a diameter of 16 units as shown. Use the
Pythagorean Theorem to determine if point H (5, √38 ) lies on circle D, and then explain your
reasoning.
7. Consider circle G with its center point located at (3, 0)
and point M (3, 2) on the circle. Determine whether
each point lies on circle G, and then explain your
reasoning.
a. 𝐽 (4.5,
√3
2
)
b. P (4, √3)
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Secondary 2
8. Write the equation of the circle
Chapter 18
𝑥 2 + 𝑦 2 + 6𝑥 − 2𝑦 + 1 = 0 in standard form. Then
identify the center point and radius of the circle.
9. Determine if the equation 81𝑥 2 + 81𝑦 2 + 36𝑥 − 324𝑦 + 327 = 0 represents a circle. If so,
describe the location of the center and radius.
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Secondary 2
Chapter 18
Additional Notes
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Secondary 2
Chapter 18
18.4 & 18.5 – Equation of a Parabola & More
(Standard: G.GPE.2)
You previously studied parabolas as quadratic functions. You analyzed equations and graphed
parabolas based on the position of the vertex and additional points determined by using x-values on
either side of the axis of symmetry. In this lesson, you will explore a parabola as a locus of points. A
locus of points is a set of points that share a property.
A parabola is the set of all points in a plane that are equidistant from a fixed point and a fixed line.
The focus of a parabola is the fixed point.
The directrix of a parabola is the fixed line.
1. The center of the circle is represented by point X. The radius of the smallest circle with
center X is 1 unit with the radius of each successive circle increasing by 1 unit.
a. What is the distance from the star to
point X, the focus? How do you
know?
b. What is the distance from the star to
the given line, the directrix? How do
you know?
c. What is the relationship between the
distance from the focus to the star
and the distance from the directrix to
the star?
d. Graph 6 additional points such that the distance from a point to the focus is equal to the
distance from the point to the directrix.
e. Draw a parabola by connecting the points with a smooth curve.
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Secondary 2
Chapter 18
2. A parabola is defined such that all points on the parabola are equidistant from the point
(0, 2) and the line 𝑦 = −2. One point on the parabola is labeled as (x, y). Determine the
equation of the parabola by completing the steps.
a. Let 𝑑1 represent the distance from (x, y) to
(0, 2). Write an equation using the Distance
Formula to represent 𝑑1 . Simplify the
equation.
b. Let 𝑑2 represent the distance from (x, y) to
the line 𝑦 = −2. Write an equation using the
Distance Formula to represent 𝑑2 . Simplify
the equation.
c.
What do you know about the relationship between 𝑑1 and 𝑑2 ?
d. Write an equation for the parabola using Question 1, parts (a) through (c). Simplify the
equation so that one side of the equation is 𝑥 2 .
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Secondary 2
Chapter 18
The general form of a parabola centered at the origin is an equation of the form:
𝐴𝑥 2 + 𝐷𝑦 = 0 𝑜𝑟 𝐵𝑦 2 + 𝐶𝑥 = 0
The standard form of a parabola is an equation in the form:
(𝑥 − ℎ)2 = 4𝑝(𝑦 − 𝑘)
𝑜𝑟
(𝑦 − 𝑘)2 = 4𝑝(𝑥 − ℎ),
where p represents the distance from the vertex to the focus and (h, k) is the vertex.
3. Write the equation of the parabola from Question 2 in general form and in standard form.
a. What are the coordinates for the x-intercept(s)
of the parabola?
b. What are the coordinates for the y-intercept(s)
of the parabola?
c. How many points on the parabola have an
x-coordinate of 4? Calculate the coordinates of
each point.
d. How many points on the parabola have a y-coordinate of 4 .5? Calculate the coordinates
of each point
e. Sketch the parabola on the grid shown using the points from Question 3a – d.
f. What is the focus? The directrix? Graph both on the coordinate plane.
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Secondary 2
Chapter 18
The axis of symmetry of a parabola is a line that passes through the parabola and divides the
parabola into two symmetrical parts that are mirror images of each other.
The vertex of a parabola is a maximum or minimum point on the curve.
The concavity of a parabola describes the orientation of the curvature of the parabola.
A parabola can be concave up, concave down, concave right, or concave left, as shown.
4. Consider the parabola represented by the equation (𝑦 + 1)2 = 12(𝑥 + 3).
a. Determine the following
x-intercept(s) :
y-intercept(s)
Coordinates when x=8:
Axis of symmetry:
Vertex:
Value of P:
b. Sketch the parabola using the coordinates
from parts (a)
c. Is the axis of symmetry along the x-axis or
the y-axis?
d. Describe the concavity of the parabola.
e. Give the focus and directrix. Place them on
the graph.
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Secondary 2
Chapter 18
5. Consider the parabola represented by the equation 𝑥 2 = 9𝑦.
a. Determine the following
x-intercept(s) :
y-intercept(s)
Coordinates when y=4:
Axis of symmetry:
Vertex:
Value of P:
b. Sketch the parabola using the coordinates
from parts (a)
c. Is the axis of symmetry along the x-axis or
the y-axis?
d. Describe the concavity of the parabola.
e. Give the focus.
f. Give the directrix.
g. Place the focus and directrix on the
coordinate plane.
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Chapter 18
6. Determine the following for each parabola:
• The vertex
• The axis of symmetry
• The value of p
• The focus
• The directrix
• The concavity
Then, graph the parabola. Label the vertex, the axis of symmetry, the focus, and the
directrix.
a. 𝑦 2 = 20𝑥
b. 𝑥 2 = −12𝑦.
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Secondary 2
Chapter 18
7. The main cables of a suspension bridge are parabolic. The parabolic shape allows the cables
to bear the weight of the bridge evenly. The distance between the towers is 900 feet and the
height of each tower is about 75 feet.
Write an equation for the parabola that represents the cable between the two towers.
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Secondary 2
Chapter 18
Parabola
Graph
Equation of
Parabola
Orientation of
Parabola
Axis of
Symmetry
Coordinates of
Vertex
Coordinates of
Focus
Equation of
Directrix
Concavity
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Secondary 2
Chapter 18
Additional Notes
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