This document provides instructions on how to sketch parabolas by identifying their key features: 1) Roots occur where the curve cuts the x-axis (where y = 0) 2) Turning points can be found by completing the square of the equation 3) The y-intercept occurs where x = 0 It demonstrates finding these features for equations such as y = x^2 - 2x - 15, identifying the minimum turning point as (1, -16), roots as (5, 0) and (-3, 0), and y-intercept as (0, -15). Finally, it asks the key question of sketching y = x^2 - 6x + 5, showing its

1. Block 3
Sketching Parabolas

2. What is to be Learned?
• How to sketch a parabola
– Showing roots
– And Turning points
Not Yet!

3. Turning Points
y = x2
– 2x – 15
Complete the square
= (x – 1)2
– 1 – 15
= (x – 1)2
– 16
Min Value of y = - 16, when x = 1
i.e Min Turning Point (1 , -16)

4. Very Handy
For y = x2
– 2x – 15
Must be
Minimum Turning Point
(1 , -16)

5. y = x2
+ 6x + 4
Complete the square
= (x + 3)2
– 9 + 4
= (x + 3)2
– 5
Min Value of y = -5, when x = -3
i.e Min Turning Point (-3, -5)
Minimum Turning Point
(-3, -5)

6. Roots
Occur where curve cuts x axis
Where y = 0
Roots
Turning
Point

7. Minimum Turning Point
(1 , -16)
y = x2
– 2x – 15

8. Finding Roots
y = x2
– 2x – 15
For Roots y = 0
= x2
– 2x – 15
or x2
– 2x – 15 = 0
(x – 5)(x + 3) = 0
x – 5 = 0 or x + 3 = 0
x = 5 x = -3
Coordinates of Roots (5 , 0) and (-3 , 0)
y0
(1 , -16)
y = x2
– 2x – 15

9. Finding Roots
y = x2
– 4x
For Roots y = 0
= x2
– 4x
or x2
– 4x = 0
x(x – 4) = 0
x = 0 or x – 4 = 0
x = 0 x = 4
Coordinates of Roots (0 , 0) and (4 , 0)
y0

10. Finding Y Intercept
Very Easy
Occurs where x = 0
y = x2
– 2x – 15
X= 0 y = 02
– 2(0) – 15
= -15
Y Intercept (0 , -15)

11. Sketching Parabolas
Vital Points and Tactics
• Roots (y = 0)
• Y Intercept (x = 0)
• Turning point (complete square)

12. Ex y = x2
+ 6x + 8
Roots (y = 0)
x2
+ 6x + 8 = 0
Factorising (x + 4)(x + 2) = 0
x + 4 =0 or x + 2 =0
x = -4 or x = -2
Roots (-4 , 0) and (-2 , 0)
Y Intercept (x = 0)
v = 02
+ 6(0) + 8
= 8
Y intercept (0 , 8)

13. Turning point (complete Square)
y = x2
+ 6x + 8
= (x + 3)2
– 9 + 8
= (x + 3)2
– 1
Min Value, y = -1, when x = -3
Min Turning Point (-3, -1)
Shape Positive

14. x
y
8
-4 -2
(-3, -1)

15. x
y
6
1 5
(3, -4)
Key Question
Sketch y =x2
– 6x + 5
showing turning point, roots
and y intercept