Question

Compare the function ƒ(x) = –x2 + 4x – 5 and the function g(x), whose graph is shown. Which function has a greater absolute maximum (vertex)? Question 4 options: A) g(x) B) g(x) and ƒ(x) have equal absolute maximums. C) ƒ(x) D) There isn't enough information given.

Answer 1

g(x) has a greater absolute maximum.

Answer 2

A) g(x) has a greater absolute maximum.

Explanation:

Given graph of g(x) which is a Parabola

1. Opens downwards

2. The absolute maximum (vertex) is at around (3.5, 6)

i.e. value of absolute maximum is 6.

Another function:

`f(x) =-x^(2)+4x-5`

Let us convert it to vertex form to find its vertex.

Taking - sign common:

`f(x) =-(x^(2)-4x+5)`

Now, let us try to make it a whole square,

Writing 5 as 4+1:

`f(x) =-(x^(2)-4x+4+1)\\\Rightarrow f(x) =-((x^(2)-2 * 2* x+2^2)+1)\\\Rightarrow f(x) =-((x-2)^(2)+1)\\\Rightarrow f(x) =-(x-2)^(2)-1`

Please refer to attached graph of f(x).

We know that, vertex form of a parabola is given as:

`f (x) = a(x - h)^2 + k`

Comparing the equations we get:

a = -1 (Negative value of a means the parabola opens downwards)

h = 2, k = -1

Vertex of f(x) is at (2, -1) i.e. value of absolute maximum is -1

and

Vertex of g(x) is at (3.5, 6)

i.e. value of absolute maximum is 6.

Hence, correct answer is:

A) g(x) has a greater absolute maximum.