Question

OMG PLEASE HELP!

The function f(x) = 4(x-7)^2-22 has been rewritten using the completing-the-square method. The function g(x) = 6x^2+12x+14 . Which vertex for each function has a minimum or a maximum? What is the vertex for each function? Explain your reasoning for each function

Answer 1

Answers:

Both functions have minima at the vertex. The vertices are (7,-22) and (-1, 8) respectively.

Explanation:

Hello! Let's look at the general form for the vertex equation of a parabola:

y-k = a(x-h)^2, or (alternativelyl) y = a(x-h)^2 + k, where (h,k) is the vertex.

Comparing the given f(x) to y = a(x-h)^2 + k, we see immediately that h = 7 and k = -22, so that the vertex is (7,-22). Since that coefficient 4 is positive, we know that the graph of this parabola opens up and that the vertex represents the minimum of f(x).

We could complete the square to rewrite g(x) in vertex form, or we could recall that the equation of the axis of symmetry is x = -b/ (2a). Here, a = 6 and b = 12, so the axis of symmetry is x = -12 / (2*6), or x = -1. Using synthetic division to evaluate g(x) at this x = -1, we get y = g(-1) = 8. Thus, the vertex of g(x) is (-1, 8). This represents the minimum value of g(x).

Both functions have minima at the vertex.

Answer 2

Explanation:

If we write a quadratic in vertex form:

y

=

a

(

x

−

h

)

2

+

k

Then:

a

8888

is the coefficient of

x

2

h

8888

is the axis of symmetry.

k

8888

is the max/min value of the function.

Also:

If

a

>

0

then the parabola will be of the form

⋃

and will have a minimum value.

If

a

<

0

then the parabola will be of the form

⋂

and will have a maximum value.

For the given functions:

a

<

0

f

(

x

)

=

−

(

x

−

1

)

2

+

5

8888

this has a maximum value of

5

a

>

0

f

(

x

)

=

(

x

+

2

)

2

−

3

8888888

this has a minimum value of

−

3

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