Math 120 - Cooleytrigonometry OCC

$Math 120 - Cooleytrigonometry OCC$

Math 120 - CooleyTrigonometry OCC

Section 1.2 – Intercepts; Symmetry; Graphing Key Equations; Circles

Computing the x and y–intercepts
To Find the x–intercept: / To Find the y–intercept:
Let y = 0, and solve for x. / Let x = 0, and solve for y.
Note: x–intercepts will always look like  (, 0) / Note: y–intercepts will always look like  (0, )
Symmetry with
respect to the x–axis / Symmetry with respect to the y–axis / Symmetry with respect to the origin
Graphical Condition / If (x, y) is on the graph, then (x, –y) is on the graph. / If (x, y) is on the graph, then (–x, y) is on the graph. / If (x, y) is on the graph, then (–x, –y) is on the graph.
Graphical Interpretation / x–axis acts a as mirror
(reflection in the x–axis) / y–axis acts as a mirror
(reflection in the y–axis) / 1) reflection in the y-axis,
2) followed by reflection inthe x–axisor visa versa.
Test Condition / Replace y with –y / Replace x with –x / Replace x with –x , AND
y with –y
If an equivalent equation results, the graph has the desired symmetry.
Hint  Think OPPOSITE
Symmetry with respect
to the x–axis / Symmetry with respect
to the y–axis / Symmetry with respect
to the origin

Definition

The standard form of an equation of a circle with radius r and center (h , k) is .

Theorem

The standard form of an equation of a circle of radius r with center at the origin (0 , 0) is .

Definition

If the radius r = 1, the circle whose center is at the origin is called the unit circle and has the equation .

Definition

The general form of the equation of the circle .

 Exercises

1)Complete the graph so that it has the type of symmetry indicated.

x–axis / y–axis / Origin

2)List the intercepts and test for symmetry:

3)List the intercepts and test for symmetry:

4)List the intercepts and test for symmetry:

5)Write the standard form of the equation and the general form of the equation of each circle of radius 4

and center (2 , –3). Then graph the circle.

6)Write the standard form of the equation and the general form of the equation of each circle of radius 7

and center (–5 , –2). Then graph the circle.

 Exercises

7)Find the center and radius of the equation of a circle:

8)Find the center and radius of the equation of a circle:

9)Find the center and radius of the equation of a circle:

10)Find the center and radius of the equation of a circle:

11)Find the standard form of the equation of the circle with endpoints of a diameter at (4 , 3) and (0 , 1).

12)Find the standard form of the equation of the circle with center (4 , –2) and tangent to theline x = 1.