Distance, Circles, And Quadratic Equations Worksheets Page 5
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9Appendix H: Distance, Circles, and Quadratic Equations
H5
√
•
(k > 0) The graph is a circle with center (x
) and radius
, y
k.
0
0
(k = 0) The only solution of the equation is x = x
, y = y
•
, so the graph is the single
0
0
point (x
, y
).
0
0
•
(k < 0) The equation has no real solutions and consequently no graph.
Example 8
Describe the graphs of
(a) (x − 1)
+ (y + 4)
= −9
(b) (x − 1)
+ (y + 4)
= 0
2
2
2
2
Solution (a).
There are no real values of x and y that will make the left side of the equa-
tion negative. Thus, the solution set of the equation is empty, and the equation has no graph.
Solution (b).
The only values of x and y that will make the left side of the equation 0
are x = 1, y = −4. Thus, the graph of the equation is the single point (1, −4).
The following theorem summarizes our observations.
H.3 theorem
An equation of the form
The last two cases in Theorem H.3 are
called
degenerate cases
. In spite of
2
+ Ay
2
+ Dx + Ey + F = 0
(6)
the fact that these degenerate cases
Ax
can occur, (6) is often called the
where A = 0, represents a circle, or a point, or else has no graph.
general equation of a circle
.
2
THE GRAPH of y = ax
+ bx + c
An equation of the form
y = ax
+ bx + c (a = 0)
2
(7)
is called a quadratic equation in x. Depending on whether a is positive or negative, the
graph, which is called a parabola, has one of the two forms shown in Figure H.7. In both
cases the parabola is symmetric about a vertical line parallel to the y-axis. This line of
symmetry cuts the parabola at a point called the vertex. The vertex is the low point on the
curve if a > 0 and the high point if a < 0.
y
y
Vertex
Vertex
x
x
−b
−b
/
/
(2a)
(2a)
y = ax
+ bx + c
y = ax
+ bx + c
2
2
a > 0
a < 0
Figure H.7
In the exercises (Exercise 78) we will help the reader show that the x-coordinate of the
vertex is given by the formula
b
x = −
(8)
2a
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