Question

g(x) = -(x - 5)^2 + 9Function fis represented by the graph, and function g is represented by the equation. Compare the functions, and choosethe statements that are true about the extreme value and axis of symmetry of the functions.

Answer 1

Step-by-step explanation:

For ease of clarity, let us write g(x) in the standard form

We have this as:

`\begin{gathered} g(x)=-(x-5)^2\text{ + 9} \\ g(x)=-(x^2-10x+25)+9 \\ g(x)=-x^2+10x-25+9 \\ g(x)=-x^2+10x-16\text{ } \end{gathered}`

So, now let us get the extreme values and axis of symmetry for both functions

Let us start with g(x)

For g(x), we have the vertex at (5,9)

For g(x), we have the axis of symmetry as

x = -b/2a

where a is the coefficient of x^2 and b is the coefficent of x (a is -1 and b is 10)

So we have the axis of symmetry for g(x) as x = -(10)/(2(-1) = 5 : x = 5

For the extreme value, we have this at (5,9)

f(10) = -(10)^2 + 10(10) -16 = -16

Now, let us move to f(x)

For f(x), the extreme value is at the vertex, which is 16 (vertex is at the point (2,16))

The axis of symmetry is the line x = 2

Now, let us select the correct option here:

We look at the options one after the other:

a) This is incorrect

b) Both functions have no minimum but maximum due to the shape of the parabola

c) This is correct (16 is greater than 9)

d) This is incorrect

e) This is correct

f) This is incorrect

The third and the fifth options are the correct options