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Mathematics Magazine for Grades 1-12  

 

Grade 11

 
Theory:

 The Conic Sections. For any of the below with a center (j, k) instead of (0, 0), replace each x term with (x-j) and each y term with (y-k).

 

Circle

Ellipse

Parabola

Hyperbola

Equation (horiz. vertex):

x2 + y2 = r2

x2 / a2 + y2 / b2 = 1

4px = y2

x2 / a2 - y2 / b2 = 1

Equations of Asymptotes:

 

 

 

y = ± (b/a)x

Equation (vert. vertex):

x2 + y2 = r2

y2 / a2 + x2 / b2 = 1

4py = x2

y2 / a2 - x2 / b2 = 1

Equations of Asymptotes:

 

 

 

x = ± (b/a)y

Variables:

r = circle radius

a = major radius (= 1/2 length major axis)
b = minor radius (= 1/2 length minor axis)
c = distance center to focus

p = distance from vertex to focus (or directrix)

a = 1/2 length major axis
b = 1/2 length minor axis
c = distance center to focus

Eccentricity:

0

c/a

1

c/a

Relation to Focus:

p = 0

a2 - b2 = c2

p = p

a2 + b2 = c2

Definition: is the locus of all points which meet the condition...

distance to the origin is constant

sum of distances to each focus is constant

distance to focus = distance to directrix

difference between distances to each foci is constant

  Problems with Solutions:

1.        Find the twentieth term and the sum of the first 20 terms of the arithmetic progression 4, 9, 14, 19….

Solution: For this progression a = 4, d = 5, and n = 20; The twentieth term is l = a + (n - 1) ∙ d = 4  + 19 ∙ 5 = 99. And the sum of the first 20 terms is: = 1030

 

2.        Insert five arithmetic means between 4 and 22.

Solution: We have a = 4, l = 22, and n = 5 + 2 = 7. Then 22 = 4 + 6 ∙ d and d = 3. The first mean is 4 + 3 = 7, the second is 7 + 3 = 10, and so on. The required means are: 7, 10, 13, 16, 19 and the resulting progression is 4, 7, 10, 13, 16 19, 22

 

3.        Find the arithmetic mean of the two numbers a and l.

Solution: We seek the middle term of an arithmetic progression of three terms having a and l as first and third terms, respectively. If d is the common difference, then a + d = l – d and d = (l - a). The arithmetic mean is a + d = a + (l - a) = (l + a).

 

4.        Find the ninth term and the sum of the first mine terms of the geometric progression 8, 4, 2, 1,….

Solution:  Here a = 8, and r = ½, and n = 9; The ninth term is

l = arn-1= The sum of the first nine terms is:

Proposed Problems:

1.        Graph each function and state its domain and range. y = 3x2 + 4

2.        For the following parabola find:

i) the direction of opening
ii) the coordinates of the vertex
iii) the y-intercept
iv) the x-intercepts

y = x2 + 3

 

3.        Find the equation of each parabola vertex at (0, -2) and passing through the point (3,7)

 

Read more on the written version of the publication.