Question

Graph the function using its vertex, axis of symmetry, and intercepts. f (x) = x2 – 12x + 36

Answer 1

Answer:

To graph a quadratic equation we first need to factorise it into a different form.

First we check what the discriminant is equal to

Where

f

(

x

)

=

a

x

2

+

b

x

2

+

c

Δ

(Discriminant)

=

b

2

−

4

a

c

In this case

Δ

=

3

2

−

4

⋅

2

⋅

7

Δ

=

−

47

Because it is less than zero it can't be factored normally

Therefore we must use the The Quadratic Formula or Completing the Square

Here I have completed the square

f

(

x

)

=

2

x

2

−

3

x

+

7

Remove factor from

x

2

term

f

(

x

)

=

2

⋅

(

x

2

−

3

2

x

+

7

2

)

Take

x

term, half it and then square it

f

(

x

)

=

−

3

2

→

−

3

4

→

9

16

Add and then subtract this number inside the equation

f

(

x

)

=

2

⋅

(

x

2

−

3

2

x

+

9

16

−

9

16

+

7

2

)

Combine the first three terms in a perfect square

f

(

x

)

=

2

⋅

(

(

x

−

3

4

)

2

−

9

16

+

7

2

)

Equate left over terms

f

(

x

)

=

2

⋅

(

(

x

−

3

4

)

2

+

47

16

)

Multiply coefficient back in

f

(

x

)

=

2

(

x

−

3

4

)

2

+

47

8

This gives a turning point of

(

3

4

,

47

8

)

=

(

0.75

,

5.875

)

and a

y

intercept of

2

⋅

(

3

4

)

2

+

47

8

=

(

0

,

7

)

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