Answer:
Part 1) Option A. The graph of g(x) has the lesser y-intercept
Part 2) Option B. (2,4)
Part 3) Option B. 62.5 miles per hour
Part 4) Option A. The functions will intersect at x=-2 and at x=1
Step-by-step explanation:
Part 1)
Let
x ----->the number of years
g(x) ----> the expected value in dollars of televisions
we know that
The linear equation in slope intercept form is equal to
[tex]y=mx+b[/tex]
where
m is the slope or unit rate
b is the y-intercept or initial value
In this problem
In the function g(x)
The slope is
[tex]m=-\$90\ per\ year[/tex] ---> is negative because is a decreasing function
The y-intercept is
[tex]b=\$360[/tex]
substitute
[tex]g(x)=-90x+360[/tex]
Find the x-intercept (value of x when the value of the function is equal to zero)
For g(x)=0
[tex]0=-90x+360[/tex]
[tex]x=4[/tex]
The x-intercept is the point (4,0)
The y-intercept is the point (0,360)
Function f(x)
we have
[tex]f(x)=100(-x+4)[/tex]
[tex]f(x)=-100x+400[/tex]
Find the x-intercept (value of x when the value of the function is equal to zero)
For f(x)=0
[tex]0=-100x+400[/tex]
[tex]x=4[/tex]
The x-intercept is the point (4,0)
The y-intercept is the point (0,400) (value of f(x) when the value of x is zero)
Compare the intercepts both functions
The y-intercept is the same in both functions
The y-intercept is greater in the function f(x)
therefore
The graph of g(x) has the lesser y-intercept
Part 2) we have the equation of the line
[tex]4x-3y=-4[/tex]
Remember that
If a ordered pair is on the given line, then the ordered pair must satisfy the given line
Verify each case
a) (4,4)
Substitute the value of x and the value of y in the equation and compare the result
[tex]4(4)-3(4)=-4[/tex]
[tex]4=-4[/tex] ---> is not true
so
the ordered pair not satisfy the equation
therefore
the ordered pair is not on the line
b) (2,4)
Substitute the value of x and the value of y in the equation and compare the result
[tex]4(2)-3(4)=-4[/tex]
[tex]-4=-4[/tex] ---> is true
so
the ordered pair satisfy the equation
therefore
the ordered pair is on the line
c) (2,-4)
Substitute the value of x and the value of y in the equation and compare the result
[tex]4(2)-3(-4)=-4[/tex]
[tex]20=-4[/tex] ---> is not true
so
the ordered pair not satisfy the equation
therefore
the ordered pair is not on the line
d) (4,-4)
Substitute the value of x and the value of y in the equation and compare the result
[tex]4(4)-3(-4)=-4[/tex]
[tex]28=-4[/tex] ---> is not true
so
the ordered pair not satisfy the equation
therefore
the ordered pair is not on the line
Part 3) we know that
The formula to calculate the average rate of change using the graph is equal to
[tex]\frac{f(b)-f(a)}{b-a}[/tex]
In this problem we have
[tex]f(a)=f(2)=75\ mi[/tex]
[tex]f(b)=f(4)=200\ mi[/tex]
[tex]a=2\ h[/tex]
[tex]b=4\ h[/tex]
Substitute
[tex]\frac{200-75}{4-2}=125/2=62.5 mi/h[/tex]
Part 4) we know that
The solution of the equation
f(x)=g(x)
are the x-coordinates of the intersection points both graphs
In this problem
If the solutions are x=-2 and x=1
then
The functions will intersect at x=-2 and at x=1
