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AP Precalculus
Unit 4: Functions Involving Parameters, Vectors, and Matrices
4.2 Graphing conic sections using implicitly defined functions and parametric functions
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Implicitly defined functions can describe curved shapes like circles and
parabolas
.
True
The focal length is a key parameter for the
parabola
What method is used to convert general conic equations to standard forms?
Completing the square
The hyperbola equation includes both addition and subtraction of
x
2
x^{2}
x
2
and y^{2}</latex> terms
False
The standard form of a circle is
(x - h)^{2} + (y - k)^{2} = r^{2}
The center of a hyperbola is denoted by
(h, k)
From which form of a conic section equation can key parameters be extracted?
Standard form
The general equation of an ellipse is
Ax^{2} + Cy^{2} + Dx + Ey + F = 0
What is the standard form of an ellipse?
(
x
−
h
)
2
a
2
+
\frac{(x - h)^{2}}{a^{2}} +
a
2
(
x
−
h
)
2
+
(
y
−
k
)
2
b
2
=
\frac{(y - k)^{2}}{b^{2}} =
b
2
(
y
−
k
)
2
=
1
1
1
What are the key parameters of an ellipse?
Center, Semi-major axis, Semi-minor axis
Completing the square is necessary to transform a general equation into
standard
form.
The semi-major axis is a key parameter of a
hyperbola
.
True
The vertex of a parabola is always
(
h
,
k
)
(h, k)
(
h
,
k
)
in its standard form.
True
What is an implicitly defined function?
Equation involving x and y
Match the conic section with its general equation:
Circle ↔️
A
x
2
+
Ax^{2} +
A
x
2
+
A
y
2
+
Ay^{2} +
A
y
2
+
D
x
+
Dx +
D
x
+
E
y
+
Ey +
E
y
+
F
=
F =
F
=
0
0
0
Ellipse ↔️
A
x
2
+
Ax^{2} +
A
x
2
+
C
y
2
+
Cy^{2} +
C
y
2
+
D
x
+
Dx +
D
x
+
E
y
+
Ey +
E
y
+
F
=
F =
F
=
0
0
0
Parabola ↔️
A
x
2
+
Ax^{2} +
A
x
2
+
D
x
+
Dx +
D
x
+
E
y
+
Ey +
E
y
+
F
=
F =
F
=
0
0
0
Hyperbola ↔️
A
x
2
−
C
y
2
+
Ax^{2} - Cy^{2} +
A
x
2
−
C
y
2
+
D
x
+
Dx +
D
x
+
E
y
+
Ey +
E
y
+
F
=
F =
F
=
0
0
0
Conic sections can be described using both implicit and parametric equations.
True
What is the term for the minor axis of an ellipse?
Semi-minor axis
Match the conic section with its identifying feature:
Circle ↔️ Same coefficients for
x
2
x^{2}
x
2
and
y
2
y^{2}
y
2
Parabola ↔️ Only one squared term
Hyperbola ↔️ Opposite signs for
x
2
x^{2}
x
2
and
y
2
y^{2}
y
2
The semi-major axis of an ellipse is denoted by 'b'
False
If the coefficients of
x
2
x^{2}
x
2
and
y
2
y^{2}
y
2
in a general conic section equation have the same sign, it's a circle or ellipse
True
If the coefficients of
x
2
x^{2}
x
2
and
y
2
y^{2}
y
2
have the same sign, the conic section is either a circle or an ellipse
Match the conic section with its key parameters:
Circle ↔️ Center, Radius
Ellipse ↔️ Center, Semi-major axis
Parabola ↔️ Vertex, Focal length
Hyperbola ↔️ Center, Semi-minor axis
What type of shapes are implicitly defined functions commonly used to describe?
Curved shapes
Conic sections are formed by intersecting a plane with a
double cone
.
True
Match the conic section with its general equation:
Circle ↔️
A
x
2
+
Ax^{2} +
A
x
2
+
A
y
2
+
Ay^{2} +
A
y
2
+
D
x
+
Dx +
D
x
+
E
y
+
Ey +
E
y
+
F
=
F =
F
=
0
0
0
Ellipse ↔️
A
x
2
+
Ax^{2} +
A
x
2
+
C
y
2
+
Cy^{2} +
C
y
2
+
D
x
+
Dx +
D
x
+
E
y
+
Ey +
E
y
+
F
=
F =
F
=
0
0
0
Parabola ↔️
A
x
2
+
Ax^{2} +
A
x
2
+
D
x
+
Dx +
D
x
+
E
y
+
Ey +
E
y
+
F
=
F =
F
=
0
0
0
or
C
y
2
+
Cy^{2} +
C
y
2
+
D
x
+
Dx +
D
x
+
E
y
+
Ey +
E
y
+
F
=
F =
F
=
0
0
0
Hyperbola ↔️
A
x
2
−
C
y
2
+
Ax^{2} - Cy^{2} +
A
x
2
−
C
y
2
+
D
x
+
Dx +
D
x
+
E
y
+
Ey +
E
y
+
F
=
F =
F
=
0
0
0
or
−
A
x
2
+
- Ax^{2} +
−
A
x
2
+
C
y
2
+
Cy^{2} +
C
y
2
+
D
x
+
Dx +
D
x
+
E
y
+
Ey +
E
y
+
F
=
F =
F
=
0
0
0
The general equation of a hyperbola is
A
x
2
−
C
y
2
+
Ax^{2} - Cy^{2} +
A
x
2
−
C
y
2
+
D
x
+
Dx +
D
x
+
E
y
+
Ey +
E
y
+
F
=
F =
F
=
0
0
0
The general equation of a hyperbola can be written as
\frac{(x - h)^{2}}{a^{2}} - \frac{(y - k)^{2}}{b^{2}} = 1</latex>
How do you identify a conic section from its general equation?
By looking at coefficients
What is the focal length of a parabola?
p
p
p
To identify a conic section, examine the coefficients of the
x
2
x^{2}
x
2
and
y
2
y^{2}
y
2
terms.
What is the center of a circle in its standard form?
(
h
,
k
)
(h, k)
(
h
,
k
)
If the coefficients of
x
2
x^{2}
x
2
and y^{2}</latex> in the general equation have the same sign, the conic section is either a circle or an ellipse.
What is the focal length of a parabola?
p
p
p
Steps to determine key features from implicit conic section equations:
1️⃣ Identify the conic section
2️⃣ Complete the square
3️⃣ Rewrite in standard form
4️⃣ Extract the key parameters
What type of conic section is formed if only one of the
x
2
x^{2}
x
2
or
y
2
y^{2}
y
2
terms is non-zero?
Parabola
The focal length of a hyperbola is denoted by
p
p
p
.
False
If only one of the coefficients of the
x
2
x^{2}
x
2
and
y
2
y^{2}
y
2
terms in a conic section equation is non-zero, it's a parabola
Match the conic section with its key parameters:
Circle ↔️ Center, Radius
Ellipse ↔️ Center, Semi-major axis
Parabola ↔️ Vertex, Focal length
If the
x
2
x^{2}
x
2
and
y
2
y^{2}
y
2
coefficients in a conic section equation are equal, it represents a circle
True
The key parameters of an ellipse are its center, semi-major axis, and semi-minor
axis
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