Pure Math 30: Explained!

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2 Conics Lesson Part I - Circles Circles: The standard form of a circle is given by the equation (x - h) +(y - k) = r, where (h, k) is the centre of the circle and r is the radius. Example : Given the following graph, write the equation. The first thing you should do when given a circle is write down the coordinates of the centre. In this case, the centre is at (-,0). Next, determine the radius, which is units. Finally, plug the h, k, and r values into the standard form equation and you ll have the equation of the graph! (x-h) +(y-k) =r (x -(-)) +(y - 0) = (x + ) + y = 6 Example : Sketch the graph of and range. (x - 3) + (y + ) = 9 and state the domain To draw the graph of a circle from a standard form equation, first draw a dot at the centre of the circle. The radius can be found by taking the square root of the number on the right side. (Remember, you re given r and you just want r.) (x - 3) +(y +) = 9 Quick Tip: An easy way to read off the centre is to use values for x and y that make each bracket go to zero. (x - 3) becomes zero when x = 3 (y + ) becomes zero when y = - So, the centre is at (3,-) When writing the domain & range for an enclosed shape, we use in-between notation Domain: Leftmost Value x Rightmost Value Range: Bottom Value y Top Value For the circle in this question: Domain: 0 x 6 (Read as the domain is between zero and six ) Range: -7 y - (Read as the range is between negative seven and negative one ) 5

3 Conics Lesson Part I - Circles Questions: For each of the following graphs, write the equation, then state domain & range:

4 Conics Lesson Part I - Circles Questions: For each of the following equations, draw the graph and state domain & range: 5. (x - ) +(y - 6) = 6 6. (x - ) + y = 6 7. (x + 5) + (y - 3) = 9 8. x +(y - 3) =

5 Answers: Conics Lesson Part I - Circles. (x - 3) +(y - ) = 9 Domain: 0 x 6 Range: y 7. (x + ) + y = 9 Domain: -9 x 5 Range: -7 y 7 3. x +(y+5) =36. Domain: -6 x 6 Range: - y (x - 5) +(y + ) = 6 Domain: x 9 Range: -6 y Domain: 0 x 8 Range: y 0 Domain: -6 x 0 Range: -8 y Domain: - x Range: - y 0 Domain: -0 x 0 Range: -7 y

6 Conics Lesson Part II- Ellipses (x - h) (y - k) Ellipses: The equation of an ellipse is given by + =, a b (h, k) is the centre of the ellipse. a represents the horizontal distance from the centre to the edge of the ellipse. b represents the vertical distance from the centre to the edge of the ellipse. Example : Given the following graph, find the equation of the ellipse: First identify the centre of the ellipse, which in this case is (,-). To find the a-value, count horizontally from the centre to the right edge and you will get 5. To find the b-value, count vertically from the centre to the upper edge, and you will get 3. Example : Sketch the graph of Place a point at the centre of the ellipse (-, ). The a-value is 9=3 The b-value is 5 = 5 When you put the a & b values into the equation, remember to square them! (x +) (y - ) + = 9 5 (x - ) (y +) + = 5 9 Quick Tip: What happens when both a and b are the same number? This will give you a circle. When writing the equation of an ellipse that is really a circle, you should use a circle equation instead. x y Don t write + = 9 9 Write x + y = 9 When a is bigger (the number under x) the ellipse is horizontal. When b is bigger, (the number under y), the ellipse is vertical 56

7 Conics Lesson Part II- Ellipses Questions: Given the following graphs, write the equation

8 Conics Lesson Part II- Ellipses Questions: Given the following equations, sketch the graph. (x - 5) (y +) + = (x - 3) +(y+) = x (y+) (x + 3) y + = + = x y (x +) (y - 3) + = + =

9 Answers: Conics Lesson Part II- Ellipses (x - 3) (y + ) + = 6 9 x (y-) + = (x + ) y + = (x + ) +(y-) = 9 (x - ) (y + 3) = 6 x y + = x +y = 59

10 Answers: 7. Conics Lesson Part II- Ellipses

11 Conics Lesson Part III - parabolas Parabolas: There are two different standard form equations for parabolas. Vertical parabolas are given by: y-k= a(x-h). a is the vertical stretch factor (Vertical parabolas that open down have a negative sign with the a-value, those opening up have a positive sign.) Horizontal parabolas are given by: x -h= a(y-k). a is the horizontal stretch factor. (Horizontal parabolas that open left have a negative sign with the a-value, those opening right have a positive sign.) (h, k) is the vertex of the parabola. Try to remember the following rules when it comes to standard form parabolas: If you have an x, but no y vertical parabola. If you have a y, but no x horizontal parabola. Example : Given the following graph, write the equation. First note the coordinates of the vertex: (8,0). This gives you h & k Example : Sketch the graph of y The vertex is located at the point (5,-), and it s a upside down vertical parabola. When given a parabola equation, it may be graphed in your calculator by isolating y: y =- (x-5) - To obtain the a-value, find another point on the parabola. By inspection, the point (5, ) lies on on the graph. This can now be plugged in for x & y. Take the values above and insert them into the standard form of a horizontal parabola: To obtain the final equation, x - h= a( y- k) plug in numbers for a, h, & k, 5-8 = a(-0) leaving x & y as variables. 3 = a x 8= 3y + = - (x-5) Example 3: Sketch the graph of y += (x+) The vertex is located at the point (-,-), and it s a right-side up vertical parabola. x intercepts: This time, graph the parabola using x & y intercepts instead of the calculator. (The x & y intercept method is being used in this example to illustrate an alternative to using your graphing calculator.) Set y = 0, then solve for x. y+= (x+) 0+= (x+) = (x+) 6 = (x + ) ± = x + x=-6, y intercept: Set x = 0, then solve for y. y+= (x+) y+= (0+) y+= ( ) y+= y=-3 6

12 Conics Lesson Part III - parabolas Questions: Given the following graphs, write the equation

13 Conics Lesson Part III - parabolas Questions: Isolate y and then sketch the graph: 7. x =-(y+) 8. y +=3x 9. x -= (y+) 0. y - = - (x+3). y =-(x+). x +3=(y-3) 63

14 Conics Lesson Part III - parabolas Questions: Using x & y intercepts, graph the following parabolas 3. ( ) y += x-3. x + = ( y+) ( ) x += y y += (x+) 6

15 Answers: Conics Lesson Part III - parabolas The vertex is at (0, -) A point is (, 0) x-h=a(y-k) -0=a(0-(-)) =a a=.. The vertex is at (0, -) A point is (, 0) y-k=a(x-h) 0 -(-) = a(- 0) =a y +=x x = (y+) The vertex is at (-3, 5) The vertex is at (-, -) 3.. A point is (, -3) A point is (3, 0) y-k=a(x-h) y-k=a(x-h) -3-5 = a(-(-3)) 0 -(-) = a(3 -(-)) -8 = a() =a() -8 =a 6 a=- y -5=- (x+3) =a 6 a= y += (x+) The vertex is at (-, -) A point is (-, 0) A point is (0, 3) x-h=a(y-k) x-h=a(y-k) - - = a(0 - ) 0 -(-) = a(3 -(-)) - = a = a() 5. The vertex is at (, ) 6. x - = -(y - ) a= 6 x += (y+)

16 Answers: x=-(y+) -x = (y + ) -x = (y + ) ± -x = y + y= ± -x- Conics Lesson Part III - parabolas 7. y =± -x- y +=3x y =3x - 8. y =3x - 9. y = ± (x - ) - 0. y =- (x+3) + x-= (y+) (x - ) = (y +) (x-)= (y+) ± (x - ) = y + y = ± (x - ) -. y =-(x+). x+3 y =± +3 x+3=(y-3) x+3 =(y-3) ± x+3 = (y - 3) x+3 y = ± + 3 x+3 =y

17 Conics Lesson Part III - parabolas Answers: y-intercept x-intercepts 3.. y+= ( x-3) y+= ( x-3) y+= ( 0-3) 0+= ( x-3) = x-3 ( ) ± = x - 3 x=, 5 9 y+= 5 y= =.5 Vertex (-, -) Vertex x-intercept: y-intercepts (3, -) x+= y+ x+= y+ ( ) ( ) x+= 0+ x+= x=-3 ( ) ( ) 0+= y+ ± = y + y=-3, Vertex (-, -) Vertex x-intercepts y-intercept (-, ) y+= (x+) y += (x+) x-intercept y-intercepts x+= 0+= (x+) y += (0+) ( y-) x+= ( y-) x+= ( 0-) 0+= ( y-) 6 = (x + ) y+= ( ) x+= ± =y- ± = x+ y+= x=-3 y=-, 3 x=-6, y=