Turning point

Question 1

Example 1: Use completing the square to find the minimum point (turning point). What is the line of symmetry?
y=x^2-2x+3

Answer 1

y=x^2-2x+3
x^2-2x+3
(x+a)^2+b
(x-1)^2-(-1)^2+3
(x-1)^2+2
Minimum point: (1, 2)
Minimum point is the opposite of the a value and the same as the b value. a=-1, and b=2, so it is 1 and 2
Line of symmetry =1 aka the a value

Question 2

Example 2: Sketch the graph below stating clearly its turning point, indicating the roots and turning point.
y=x^2+9x+1

Answer 2

y=x^2+9x+1
x^2+9x+1=0
(x+9/2)^2-(9/2)^2+1=0
(x+9/2)^2-81/4+1=0
(x+9/2)^2-81/4+4/4=0
(x+9/2)^2-77/4=0
(x+9/2)^2=77/4
x+9/2=plus or minus root 77/4
x=-9/2 plus or minus root 77/4
The graph should be a u shape. It should cross the x axis at the points x=-9/2-root 77/4 and x=-9/2+root 77/4. Take note it crosses the x axis at these points because y has been set equal to zero. Also, it says x=. Also, our answer was plus or minus 77/4, so we have used both plus and minus to get our roots which cross the graph at different points. As the equation is y=x^2+9x+1, then it should cross the y axis at 1. Also, its turning point is (-9/2, -77/4). We get this from (x+9/2)^2-77/4=0. Its the opposite of the a value, so it is -9/2, and the same as the a value, so it is -77/4.

Question 3

Example 3 - Write in the form a(x+b)^2+c

2x^2+8x+5

Answer 3

Write a(x+b)^2+c
2x^2+8x+5
Factorise the 2 out
2(x+4x)+5
Complete the square to the bit in brackets
x^2+4x=(x+2)^2-2^2
=(x+2)^2-4
Substitute answer back in
2((x+2)^2-4)+5
2(x+2)^2-8+5
2(x+2)^2-3