Answer:
Yes they will intersect.
Step-by-step explanation:
Given : Neil is analyzing a quadratic function f(x) and a linear function g(x).
[tex]f(x)=x^2+4x+4[/tex] and
x -1 -3 -5
g(x) 0 1 2
To find : Will they intersect?
Solution :
Given Points for g(x) are (-1,0) , (-3,1) and (-5,2).
As g is a linear function,
Then it must be in form of g(x) = mx +c
where, c is constant and m is slope.
Using slope formula,
[tex]m=\frac{y_2-y_1}{x_2-x_1}[/tex]
Here, [tex]x_1=-1 , x_2=-3 , y_1=0, y_2=1[/tex]
[tex]m=\frac{1-0}{-3-(-1)}[/tex]
[tex]m=\frac{1}{-2}[/tex]
[tex]m=-\frac{1}{2}[/tex]
The equation form is [tex]g(x) =-\frac{1}{2}x +c[/tex]
Now, Put (-5,2) point as it also satisfy the equation
[tex]2=-\frac{1}{2}(-5) +c[/tex]
[tex]2=\frac{5}{2}+c[/tex]
[tex]2-\frac{5}{2}=c[/tex]
[tex]c=\frac{-1}{2}[/tex]
The the equation of g(x) is [tex]g(x) =-\frac{1}{2}x-\frac{1}{2}[/tex]
Now, We plot the equation of f(x) and g(x) in graphing tool.
The function [tex]f(x)=x^2+4x+4[/tex] is plotted by red curve.
The function [tex]g(x)=-\frac{1}{2}x-\frac{1}{2}[/tex] is plotted by blue curve.
The intersection point of both the curve is (-3,1) and (-1.5,0.25).
Therefore, Yes the function intersect.
Refer the attached figure below.
