lucas filled 6.5 more boxes than charlotte and xin filled 8 fewer than lucas together they filled 50 boxes how many boxes did each person fill

Part A) The x-intercepts are the points (-2.5,0) and (0.5,0)

Part B) The vertex is a minimum and the coordinates are (-1,-9)

Part C) see the explanation

Step-by-step explanation:

we have

Part A: What are the x-intercepts of the graph of f(x)?

we know that

The x-intercepts are the values of x when the value of the function f(x) is equal to zero

so

For f(x)=0

Solve the quadratic equation

The formula to solve a quadratic equation of the form

is equal to

in this problem we have

so

substitute in the formula

therefore

The x-intercepts are the points (-2.5,0) and (0.5,0)

Part B: Is the vertex of the graph of f(x) going to be a maximum or minimum? What are the coordinates of the vertex?

we know that

The quadratic equation represent a vertical parabola open upward (because the leading coefficient is positive)

therefore

The vertex represent a minimum

Convert to vertex form

Factor 4

Complete the square

Rewrite as perfect squares

> equation in vertex form

The vertex is the point (-1,-9)

Part C: What are the steps you would use to graph f(x)?

To graph f(x)

I need

1) The direction of the graph (Is a vertical parabola open upward)

2) The coordinates of the vertex (-1,-9) ( in this case is a minimum)

3) The x-intercepts  (-2.5,0) and (0.5,0)

4) The y-intercept

The y-intercept is the value of f(x) when the value of x is equal to zero

so

For x=0

The y-intercept is the point (0,-5)

Plot the points and draw the parabola

see the attached figure to better understand the problem