Question

if f(x) and g(x) are quadratic functions but (f+g)(x) produces the graph below, which statement must be true? A. the leading coefficients of f(x) and g(x) are opposites B. the leading coefficients of f(x) and g(x) are opposite reciprocals C. the leading coefficients of f(x) and g(x) are the same age. D. the leading coefficients of f(x) and g(x) are reciprocals

Answer 1

A quadratic function in its generic form is:
 f (x) = ax2 + bx + c
 g (x) = dx2 + ex + f
 The sum of the functions is:
 (f + g) (x) = (a + d) x2 + (b + e) x + (c + f)
 For the function to be linear necessarily:
 a = -d
 Answer:
 A. the leading coefficients of f (x) and g (x) are opposites

Answer 2

Answer : the leading coefficients of f(x) and g(x) are opposites

if f(x) and g(x) are quadratic functions but (f+g)(x) produces the graph below

Let f(x) = ax^2 + bx +c

g(x) = dx^2 + ex + f

The graph attached is a linear graph

we need to produce linear graph by adding two functions ,

In linear graph, there is no x^2 term

To remove x^2 term, the coefficient of x^2 term should be same and of opposite sign

if d= -a  , then g(x) = -ax^2 + 2x + f

f(x) + g(x) = (a-a)x^2 + (b+e)x + c+f= (b+e)x + c+f

f(x) + g(x) becomes linear

the leading coefficients of f(x) and g(x) are opposites