Given r = 6 cos θ 2 , a. Replace r , θ by r , π − θ in the equation. Does this show that the graph of the equation is symmetric with respect to the line θ = π 2 ? b. The ordered pair − r , 2 π − θ is another representation of the point r , π − θ . Replace r , θ by − r , 2 π − θ in the equation. Does this show that the graph of the equationis symmetric with respect to the line θ = π 2 ? c. What other type of symmetry (if any) does the graph of the equation have? d. Use a graphing utility to graph the equation on a graphing utility for 0 ≤ θ ≤ 4 π .
Given r = 6 cos θ 2 , a. Replace r , θ by r , π − θ in the equation. Does this show that the graph of the equation is symmetric with respect to the line θ = π 2 ? b. The ordered pair − r , 2 π − θ is another representation of the point r , π − θ . Replace r , θ by − r , 2 π − θ in the equation. Does this show that the graph of the equationis symmetric with respect to the line θ = π 2 ? c. What other type of symmetry (if any) does the graph of the equation have? d. Use a graphing utility to graph the equation on a graphing utility for 0 ≤ θ ≤ 4 π .
Solution Summary: The author analyzes the graph of the polar equation r=6mathrmcostheta 2.
a. Replace
r
,
θ
by
r
,
π
−
θ
in the equation. Does this show that the graph of the equation is symmetric with respect to the line
θ
=
π
2
?
b. The ordered pair
−
r
,
2
π
−
θ
is another representation of the point
r
,
π
−
θ
. Replace
r
,
θ
by
−
r
,
2
π
−
θ
in the equation. Does this show that the graph of the equationis symmetric with respect to the line
θ
=
π
2
?
c. What other type of symmetry (if any) does the graph of the equation have?
d. Use a graphing utility to graph the equation on a graphing utility for
0
≤
θ
≤
4
π
.
Consider the following system of equations, Ax=b :
x+2y+3z - w = 2
2x4z2w = 3
-x+6y+17z7w = 0
-9x-2y+13z7w = -14
a. Find the solution to the system. Write it as a parametric equation. You can use a
computer to do the row reduction.
b. What is a geometric description of the solution? Explain how you know.
c. Write the solution in vector form?
d. What is the solution to the homogeneous system, Ax=0?
2. Find a matrix A with the following qualities
a. A is 3 x 3.
b. The matrix A is not lower triangular and is not upper triangular.
c. At least one value in each row is not a 1, 2,-1, -2, or 0
d. A is invertible.
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