Question

In two or more complete sentences, compare the number of x-intercepts in the graph of f(x)=x2 to the number of x-intercepts in the graph of g(x)= (x+2)2 +7. Be sure to include the transformations that occurred between the parent function f(x) and its image g(x).

Answer 1

f(x) has exactly one intercept at the origin. g(x) is transformed from f(x) by flipping it (the minus sign at the x2) and then lifting it by 7 units. This means that the resulting mountain parabola will have two x-intercepts. Mathematically, a parabola has exactly one x-intercepts when its discriminant equals 0, it has 2 intercepts for D>0.

D= b^{2} -4ac

For g(x), a=-1, b=0 and c=7, so D=28, meaning 2 intercepts!

Answer 2

to move a function up c units, add c to whole function
to move a function left c nits, add c to every x

so
f(x)=x²
when we add 2 to every x and add 7 to whole function we get
g(x)=(x+2)²+7
and it moves the function 7 units up and 2 units to the left


see x intercepts (when y=0)

f(x)=y=x²
0=x²
at x=0, 1 x intercept

g(x)=(x+2)²+7
0=(x+2)²+7
-7=(x+2)²
false, no real solutions and therfor no real x intercepts

f(x) has 1 x intercept and g(x) has zero
g(x) is f(x) but moved up 7 units and to the left 2 u nits