Answer:
Part A) [tex]h(4) -m(16)=-4[/tex]
Part B) The distance between the y-intercepts is equal to 4 units
Part C) The value of h(x) will always be greater than the value of m(x) for any value of x
Step-by-step explanation:
Part A) What is the value of h(4) -m(16)
we have
[tex]h(x)=\frac{1}{2}(x-2)^2[/tex]
For x=4
[tex]h(4)=\frac{1}{2}(4-2)^2=2[/tex]
Find the equation of the line m(x)
Find the slope
take the points
(8,2) and (12,4)
The formula to calculate the slope between two points is equal to
[tex]m=\frac{y2-y1}{x2-x1}[/tex]
substitute
[tex]m=\frac{4-2}{12-8}[/tex]
[tex]m=\frac{2}{4}[/tex]
[tex]m=\frac{1}{2}[/tex]
Find the equation of the line in slope intercept form
[tex]y=mx+b[/tex]
we have
[tex]m=\frac{1}{2}[/tex]
[tex]point\ (8,2)[/tex]
substitute
[tex]2=\frac{1}{2}(8)+b[/tex]
solve for b
[tex]b=-2[/tex]
[tex]y=\frac{1}{2}x-2[/tex]
therefore
[tex]m(x)=\frac{1}{2}x-2[/tex]
Find m(16)
[tex]m(16)=\frac{1}{2}(16)-2=6[/tex]
so
[tex]h(4) -m(16)=2-6=-4[/tex]
Part B) we know that
The y-intercept is the value of y when the value of x is equal to zero
Find the y-intercept of h(x)
For x=0
[tex]h(0)=\frac{1}{2}(0-2)^2=2[/tex]
Find the y-intercept of m(x)
For x=0
[tex]m(0)=\frac{1}{2}(0)-2=-2[/tex]
therefore
The distance between the y-intercepts is equal to 2-(-2)=4 units
Part C)
Graph both equations
[tex]h(x)=\frac{1}{2}(x-2)^2[/tex]
[tex]m(x)=\frac{1}{2}x-2[/tex]
using a graphing tool
see the attached figure
The value of h(x) will always be greater than the value of m(x) for any value of x
