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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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What are conic sections formed by?
Intersection of cone and plane
Conic sections can be defined using
implicitly defined functions
.
What is the main difference between an implicitly defined function and an explicitly defined function?
Equation solves for one variable
Match the type of function with its example:
Implicitly Defined ↔️ (x - h)^{2} + (y - k)^{2} = r^{2}</latex>
Explicitly Defined ↔️
y
=
y =
y
=
x
2
+
x^{2} +
x
2
+
3
x
−
5
3x - 5
3
x
−
5
To graph conic sections using implicitly defined functions, the first step is to identify the type of conic
section
Steps to graph conic sections using implicitly defined functions:
1️⃣ Identify the type of conic section
2️⃣ Identify the parameters
3️⃣ Use the parameters to sketch the graph
To graph a circle, you first plot the center and then draw a circle with the given
radius
.
To graph an ellipse, you first plot the center and then draw an ellipse with major axis
2a
How do you graph a parabola using its implicit equation?
Identify the vertex
Steps to graph a hyperbola using its implicit equation:
1️⃣ Plot the center (h, k)
2️⃣ Draw the hyperbola with transverse axis 2a and conjugate axis 2b
What is the implicit equation of an ellipse with center (h, k), major axis 2a, and minor axis 2b?
(
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
The graph of a parabola is determined by the sign of the parameter
a
A hyperbola has a transverse axis of length 2a and a conjugate axis of length
2b
.
What is the implicit equation of a circle with center (h, k) and radius r?
(
x
−
h
)
2
+
(x - h)^{2} +
(
x
−
h
)
2
+
(
y
−
k
)
2
=
(y - k)^{2} =
(
y
−
k
)
2
=
r
2
r^{2}
r
2
The vertex of a parabola is given by the coordinates
(h, k)
Conic sections are formed by the
intersection
of a cone and a plane.
Match the conic section type with its implicit equation:
Circle ↔️
(
x
−
h
)
2
+
(x - h)^{2} +
(
x
−
h
)
2
+
(
y
−
k
)
2
=
(y - k)^{2} =
(
y
−
k
)
2
=
r
2
r^{2}
r
2
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
Parabola ↔️
y
=
y =
y
=
a
x
2
+
ax^{2} +
a
x
2
+
b
x
+
bx +
b
x
+
c
c
c
Hyperbola ↔️
(
x
−
h
)
2
a
2
−
(
y
−
k
)
2
b
2
=
\frac{(x - h)^{2}}{a^{2}} - \frac{(y - k)^{2}}{b^{2}} =
a
2
(
x
−
h
)
2
−
b
2
(
y
−
k
)
2
=
1
1
1
What is the main difference between an implicitly defined function and an explicitly defined function?
Equation solves for one variable
The equation for a circle,
(
x
−
h
)
2
+
(x - h)^{2} +
(
x
−
h
)
2
+
(
y
−
k
)
2
=
(y - k)^{2} =
(
y
−
k
)
2
=
r
2
r^{2}
r
2
, is an example of an implicit function.
To graph a circle, you first plot the center and then draw a circle with the
radius
r.
Steps to graph a conic section using its implicit equation:
1️⃣ Identify the type of conic section
2️⃣ Identify the parameters (center, axes, radius)
3️⃣ Sketch the graph using the parameters
What are the two axes used to graph a hyperbola?
Transverse and conjugate axes
The first step in graphing conic sections is to identify the
type
What parameters are needed to sketch a circle?
Center and radius
Steps to graph conic sections
1️⃣ Identify the type of conic section
2️⃣ Identify the parameters
3️⃣ Use the parameters to sketch the graph
The implicit equation of a circle is
(
x
−
h
)
2
+
(x - h)^{2} +
(
x
−
h
)
2
+
(
y
−
k
)
2
=
(y - k)^{2} =
(
y
−
k
)
2
=
r
2
r^{2}
r
2
What are the major and minor axes of an ellipse?
2a and 2b
The second step in graphing conic sections is to identify the
parameters
What is the vertex of a parabola defined by
y
=
y =
y
=
a
x
2
+
ax^{2} +
a
x
2
+
b
x
+
bx +
b
x
+
c
c
c
?
(h, k)
The direction a
parabola
opens depends on the sign of 'a'
What are the transverse and conjugate axes of a hyperbola?
2a and 2b
Identifying the type of conic section is the first step in graphing using
implicit
equations.
Parametric functions
use a third variable, 't', to express x and y coordinates.
What are the parametric equations for a circle with center (h, k) and radius r?
x = h + r cos(t), y = k + r sin(t)
Match the conic section with its parametric equations:
Circle ↔️ x = h + r cos(t), y = k + r sin(t)
Ellipse ↔️ x = h + a cos(t), y = k + b sin(t)
Parabola ↔️ x = h + at², y = k + 2at
Hyperbola ↔️ x = h + a sec(t), y = k + b tan(t)
Steps to graph conic sections using parametric equations
1️⃣ Identify the type of conic section
2️⃣ Identify the parameters
3️⃣ Use the parametric equations to calculate (x, y) coordinates by varying t
4️⃣ Plot the points and sketch the graph
Parametric functions use a third variable 't' to express conic sections, simplifying their
graphing
Using parametric equations allows for easier plotting of conic sections by varying
't'
How are conic sections formed?
Cone and plane intersection
Match the conic section with its implicit equation:
Circle ↔️
(
x
−
h
)
2
+
(x - h)^{2} +
(
x
−
h
)
2
+
(
y
−
k
)
2
=
(y - k)^{2} =
(
y
−
k
)
2
=
r
2
r^{2}
r
2
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
Parabola ↔️
y
=
y =
y
=
a
x
2
+
ax^{2} +
a
x
2
+
b
x
+
bx +
b
x
+
c
c
c
Hyperbola ↔️
(
x
−
h
)
2
a
2
−
(
y
−
k
)
2
b
2
=
\frac{(x - h)^{2}}{a^{2}} - \frac{(y - k)^{2}}{b^{2}} =
a
2
(
x
−
h
)
2
−
b
2
(
y
−
k
)
2
=
1
1
1
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