Question 1

LEVEL 6

Find the turning point of the graph y=x^2+3x-10.

Select the correct answer from the list below:

A: (3.5,-8.25)

B: (1.5,2.5)

C: (-1.5,-12.25)

D: (7,14)

 

CORRECT ANSWER:    C: (-1.5,-12.25)

WORKED SOLUTION:

Before we can draw a sketch, we need to factorise our equation.

y=x^2+3x-10=(x+5)(x-2)

So, we can see that the graph will go through the x-axis at x=-5 and x=2. Then, because it is a positive quadratic, it will look like a smiley face.

Now, we know that because of symmetry the turning point will be between these, so we need to find the average

x=\frac{2+(-5)}{2}=\frac{-3}{2}=-1.5

This is the x coordinate of the turning point, which we put into the original equation to find the y coordinate.

y=x^2+3x-10=(-1.5)^2+3(-1.5)-10=-12.25

So, the turning point is at (-1.5,-12.25).

Completing the Square

The coefficient of x is 3, so we need to half this to go into our bracket and then subtract its square

y=x^2+3x-10=(x+1.5)^2-10-(1.5)^2=(x+1.5)^2-10-2.25=(x+1.5)^2-12.25 .

We know that the smallest value the bracket can be is 0, so if we substitute that in we get:

y=(0)^2-12.25=-12.25

However, for this to be 0, we need to set the bracket to zero:
x+1.5=0
x=-1.5

Giving us the co-ordinate (-1.5,-12.25).

Question 2

LEVEL 6

Find the turning point of the graph y=x^2-6x+8.

Select the correct answer from the list below:

A: (6,8)

B: (5,2)

C: (-6,6)

D: (3,-1)

 

CORRECT ANSWER:   D: (3,-1)

WORKED SOLUTION:

Before we can draw a sketch, we need to factorise our equation.

y=x^2-6x+8=(x-4)(x-2)

So, we can see that the graph will cross the x-axis at x=2 and x=4. Then, because it is a positive quadratic, it will look like a smiley face.

Now, we know that because of symmetry the turning point will be between these, so we need to find the average

x=\frac{4+2}{2}=\frac{6}{2}=3

This is the x coordinate of the turning point, which we put into the original equation to find the y coordinate.

y=x^2-6x+8=3^2-6(3)+8=-1

So, the turning point is at (3,-1).

Completing the Square

The coefficient of x is -6, so we need to half this to go into our bracket and then subtract its square

y=x^2-6x+8=(x-3)^2+8-(-3)^2=(x-3)^2+8-9=(x-3)^2-1 .

We know that the smallest value the bracket can be is 0, so if we substitute that in we get:

y=(0)^2-1=-1

However, for this to be 0, we need to set the bracket to zero:
x-3=0
x=3

Giving us the co-ordinate (3,-1).

Question 3

LEVEL 6

Find the turning point of the graph y=-x^2+13x-42.

Select the correct answer from the list below:

A: (6.5, 0.25)

B: (13, 42)

C: (6, 0)

D: (-6.5, -0.25)

 

CORRECT ANSWER:  A: (6.5, 0.25)

WORKED SOLUTION:

Before we can draw a sketch, we need to factorise our equation. Because there is a minus, it is helpful to pull out the minus first, and then factorise.

y=-x^2+13x-42 =-(x^2-13x+42)
y=-(x-7)(x-6)

So, we can see that the graph will cross the x-axis at x=6 and x=7.

Then, because it is a negative quadratic, it will look like a frowny face.

Now, we know that because of symmetry the turning point will be between these, so we need to find the average

x=\frac{6+7}{2}=\frac{13}{2}=6.5

This is the x coordinate of the turning point, which we put into the original equation to find the y coordinate.

y=-x^2+13x-42=-(6.5)^2+13(6.5)-42=0.25

So, the turning point is at (6.5,0.25).

Completing the Square
Because the x^2 has a negative coefficient, it is helpful to pull out the minus again first

y=-x^2+13x-42 =-(x^2-13x+42)
And now we can complete the square.

The coefficient of x is -13, so we need to half this to go into our bracket and then subtract its square

y=-(x^2-13x+42)=-((x-6.5)^2+42-6.5^2)=-((x-6.5)^2+42-42.25)=-((x-6.5)^2-0.25) .
And then we can multiply by the -1 to get:

y=-(x-6.5)^2+0.25 .

We know that the smallest value the bracket can be is 0, so if we substitute that in we get:

y=(0)^2+0.25=0.25

However, for this to be 0, we need to set the bracket to zero:
x-6.5=0
] x=6.5

Giving us the co-ordinate (6.5,0.25).

Question 4

LEVEL 6

Find the turning point of the graph y=2x^2+x-3.

Select the correct answer from the list below:

A: (-1.5, 0)

B: (-0.25,-3.125)

C: (1, 1)

D: (-2.25, -0.25)

 

CORRECT ANSWER:   B: (-0.25,-3.125)

WORKED SOLUTION:

Drawing a Sketch

Before we can draw a sketch, we need to factorise our equation.

y=2x^2+x-3 =(2x+3)(x-1)

So, we can see that the graph will cross the x-axis at x=1 and 2x+3=0\rightarrow 2x=-3\rightarrow x=-1.5. Then, because it is a positive quadratic, it will look like a smiley face.

Now, we know that because of symmetry the turning point will be between these, so we need to find the average

x=\frac{1+(-1.5)}{2}=\frac{-0.5}{2}=-0.25

This is the x coordinate of the turning point, which we put into the original equation to find the y coordinate.

y=2x^2+3x-3=2(-0.25)^2+(-0.25)-3=-2.125

So, the turning point is at (-0.25,-3.125).

Completing the Square
Because the x^2 has a coefficient of 2, it is helpful to pull this out and divide everything by 2 first

y=2x^2+x-3 =2(x^2+0.5x-1.5)
And now we can complete the square.

The coefficient of x is 0.5, so we need to half this to go into our bracket and then subtract its square

y=2(x^2+0.5x-1.5)=2((x+0.25)^2-1.5-0.25^2)= 2((x+0.25)^2-1.5-0.0625)= 2((x+0.25)^2-1.5625) .

And then we cam multiply by the 2 to get:

y=2(x+0.25)^2-3.125 .

We know that the smallest value the bracket can be is 0, so if we substitute that in we get:

y=(0)^2-3.125=-3.125

However, for this to be 0, we need to set the bracket to zero:

x+0.25=0
x=-0.25

Giving us the co-ordinate (-0.25,-3.125).

Question 5

LEVEL 6

Find the turning point of the graph y=-x^2-10x-25.

Select the correct answer from the list below:

A: (-5, 5)

B: (0, 5)

C: (-5, 0)

D: (-5, -5)

 

CORRECT ANSWER:   C: (-5, 0)

WORKED SOLUTION:

Before we can draw a sketch, we need to factorise our equation. Because there is a minus, it is helpful to pull out the minus first, and then factorise.

y=-x^2-10x-25 =-(x^2+10x+25)
y=-(x+5)(x+5)

So, we can see that the graph will only touch thex-axis at x=-5. Then, because it is a negative quadratic, it will look like a frowny face.

Now, we know that because of symmetry the turning point is the point where the graph touches the x-axis.

Thus, the turning point is (-5,0)

 

Completing the Square
Because the x^2 has a negative coefficient, it is helpful to pull out the minus again first

y=-x^2-10x-25 =-(x^2+10x+25)
And now we can complete the square.

The coefficient of x is 10, so we need to half this to go into our bracket and then subtract its square

y=-((x+5)^2+25-5^2=((x+5)^2+25-25=-((x+5)^2) .
And then we can multiply by the -1 to get:

y=-(x+5)^2 .

We know that the smallest value the bracket can be is 0, so if we substitute that in we get:

y=(0)^2=0

However, for this to be 0, we need to set the bracket to zero:
x+5=0
x=-5

Giving us the co-ordinate (-5,0).