Answer:
[tex]\displaystyle B.g(2)[/tex]
Step-by-step explanation:
we are given f(x) and the graph of g(x)
to figure out which function has the greatest value we need to figure out g(x) function first
the given graph is a parabola so g(x) has to be quadratic function
both vertex and y-intercept of the function is (0,-1)
remember,[tex]\sf \displaystyle Q_{\text{vertex}}=g(x)=a(x-h)^2+k[/tex]
vertex:(h,k)
we got from the graph (h,k)=(0,-1)
substitute the value of h and k to the vertex form:
[tex]\displaystyle g(x)=a(x-0)^2+( - 1)[/tex]
simplify:
[tex]\displaystyle g(x)=ax^2- 1[/tex]
now we need to know figure out a
to do so take (-4,-5) coordinate pair which means if x=-4 then g(x)=-5
it is helpful to figure out a
substitute the value -4 for x and -5 for g(x):
[tex]\displaystyle - 5=a( - 4)^2- 1[/tex]
simplify square:
[tex]\displaystyle - 5=16a- 1[/tex]
add 1 to both sides:
[tex]\displaystyle - 4=16a[/tex]
divide both sides by 16:
[tex]\displaystyle a = - \frac{1}{4} [/tex]
our quadratic function is
[tex]\displaystyle g(x)= - \frac{1}{4} x^2- 1[/tex]
for f(x) and g(x) substitute the values 6,0 and 2,-4 to determine which function has the greatest value
let's work with f(x)
when x is 6 then f(x)
[tex] \displaystyle \: f(6) = \frac{2}{3} \times 6 - 8[/tex]
simplify multiplication:
[tex] \displaystyle \: f(6) = 2\times 2- 8[/tex]
simplify:
[tex] \displaystyle \: f(6) = 4- 8[/tex]
simplify substraction:
[tex] \displaystyle \: f(6) = - 4[/tex]
when x is 0 then f(x)
[tex] \displaystyle \: f(0) = \frac{2}{3} \times 0 - 8[/tex]
simplify multiplication:
[tex] \displaystyle \: f(0) = - 8[/tex]
let's work with g(x) now
when x is 2 then g(x)
[tex]\displaystyle g(2)= - \frac{1}{4} \times 2^2- 1[/tex]
simplify square:
[tex]\displaystyle g(2)= - \frac{1}{4} \times 4- 1[/tex]
simplify:
[tex]\displaystyle g(2)= - 1- 1[/tex]
simplify substraction:
[tex]\displaystyle g(2)= -2[/tex]
when x is -4 then g(x)
[tex]\displaystyle g( - 4)= - \frac{1}{4} (- 4)^2- 1[/tex]
simplify square:
[tex]\displaystyle g( - 4)= - \frac{1}{4} \times 16- 1[/tex]
simplify;
[tex]\displaystyle g( - 4)= - 1 \times 4- 1[/tex]
simplify multiplication:
[tex]\displaystyle g( - 4)= - 4- 1[/tex]
simplify subtraction:
[tex]\displaystyle g( - 4)= - 5[/tex]
so
[tex] \begin{array}{c c c} g(2) > f( 6) > g( - 4) > f(0)\\ - 2 > - 4 > - 5> - 8 \end{array}[/tex]
hence,
our answer choice is [tex]\displaystyle B.g(2)[/tex]
