Given the quadratic function f(x)=3x^2-24x+49 a. use “completing the squares” to convert the quadratic function into vertex form b. state the vertex c. find the x and y intercepts d. graph the function

See below.

Step-by-step explanation:

A.  F(x)=3x^2 - 24x + 49

f(x) = 3(x^2 - 8x) + 49

f(x) = 3 [(x - 4)^2 - 16]  + 49

f(x) = 3(x - 4)^2 - 48 + 49

f(x) =  3(x - 4)^2 + 1  < Vertex form. (answer).

B.

The vertex is at (4, 1). (answer).

C.

The y-intercept occurs when x = 0:

y = 3(0- 4)^2 + 1

= 3*16 + 1

= 49

The y-intercept is at (0, 49). (answer)

The x intercepts occur when y = 0:

3(x - 4)^2 + 1 = 0

3(x - 4)^2 = -1

(x - 4)^2 = -1/3

Since there is no real square root of a negative there are no x-intercepts.

(Answer)

x = 45

y = 60

z = 75

Step-by-step explanation:

We can create three different equations with the given variables:

x + y + z = 180

y + z = 3 x

z = y + 15

then we can use this last equation to substitute for 'z" in the second equation:

y + z = 3x

y + y + 15 = 3 x

2 y + 15 = 3 x

x = 2/3 y + 5

Then we can re-write the first equation in terms of y and solve for this unknown:

2/3 y + 5 + y + y + 15 = 180

2/3 y + 2 y = 160

8/3 y = 160

y = 3 * 160 /8

y = 60

Then   x = 2/3 (60) + 5 = 45

so x = 45

and finally:   z = 60 + 15 = 75

so  z = 75