Question

Two quadratic functions are represented below.

Answer 1

To determine the maximum value of a quadratic function opening downwards, we are going to find the vertex; then the y-value of the vertex will be our maximum.
To find the vertex (h,k) (where h=x-coordinate and k=y-coordinate) of a quadratic function of the form $f(x)=a x^{2} +bx+c$ we'll use the vertex formula: $h= \frac{-b}{2a}$, and then we are going to replace that value in our original function to find k.

So, in our function $f(x)=- x^{2} +4x-3$, $a=-1$ and $b=4$. 
Lets replace those values in our vertex formula:
$h= \frac{-4}{2(-1)}$
$h= \frac{-4}{-2}$
$h=2$

Now that we know the x-coordinate of our vertex, lets replace it in the original function, to get the y-coordinate:
$h=f(2)=-(2^{2} )+4(2)-3$
$h=-4+8-3$
$h=1$

We just prove that the vertex of $f(x)$ is (2,1), and for the graph we can tell that the vertex of $g(x)$ is (-2,4). The only thing left is compare their y-coordinates to determine which one has the greater maximum value. Since 4>1, we can conclude that $g(x)$ has the greater maximum.

Answer 2

1) The first way is graphing. You can find the maximum value visually by graphing the equation and finding the maximum point on the graph

2) The second way to determine the maximum value is using the equation y = ax² + bx + c.

If your equation is in the form ax² + bx + c, you can find the maximum by using the equation:

max/min = c - (b2 / 4a).

The first step is to determine whether your equation gives a maximum or minimum. This can be done by looking at the x² term. 

If x²>0------------ > the vertex point will be a minimum. 

If x²<0-------------> the vertex will be a maximum.

3)There's one more way to determine the maximum value of a function, and that is from the equation:

y = a(x - h)2 + k

If the a term in this equation must be negative for there to be a maximum. 

If the a term is negative, the maximum can be found at k. No equation or calculation is necessary - the answer is just k.

using a graph tool

see the attached figure

the maximum value of f(x) is 1 at the point (2,1)

the maximum value of g(x) is 4 at the point (-2,4)

the answer is 

The function g(x) has the greater maximum value