Answer:
Part 1) option a. [tex]y=(x+1)^{2}[/tex]
Part 2) option c. [tex]y(x)=10x+1[/tex]
Part 3) option a. Yes , d=-2
Part 4) option b. [tex]y=2x+4[/tex]
Part 5) option b. [tex]m=-2[/tex]
Part 6) option c. [tex]y=4x+14[/tex]
Part 7) option c. [tex]y=4x+5[/tex]
Part 8) option a. y=2x-1 and y=x+1
Step-by-step explanation:
Part 1)
we know that
If a ordered pair satisfy a function, then the function pass through the ordered pair
Verify each function with the points (1,4), (2,9) and (3,16)
case a) we have
[tex]y=(x+1)^{2}[/tex]
For x=1, y=4
[tex]4=(1+1)^{2}[/tex]
[tex]4=4[/tex] ----> is true
For x=2, y=9
[tex]9=(2+1)^{2}[/tex]
[tex]9=9[/tex] ----> is true
For x=3, y=16
[tex]16=(3+1)^{2}[/tex]
[tex]16=16[/tex] ----> is true
therefore
The function pass through the three points
case b) we have
[tex]y=(x+3)^{2}[/tex]
For x=1, y=4
[tex]4=(1+3)^{2}[/tex]
[tex]4=16[/tex] ----> is not true
therefore
The function not pass through the three points
case c) we have
[tex]y=7x-5[/tex]
For x=1, y=4
[tex]4=7(1)-5[/tex]
[tex]4=2[/tex] ----> is not true
therefore
The function not pass through the three points
Part 2)
Let
y------> the number of laps
x-----> the number of hours
we know that
The linear equation that represent this situation is
[tex]y(x)=10x+1[/tex]
Part 3) we have
{4,2,0,-2,-4,-6,...}
Let
a1=-4
a2=2
a3=0
a4=-2
a5=-4
a6=-6
we know that
a2-a1=2-4=-2 -----> a2=a1-2
a3-a2=0-2=-2 ----> a3=a2-2
a4-a3=-2-0=-2 -----> a4=a3-2
a5-a4=-4-(-2)=-2----> a5=a4-2
a6-a5=-6-(-4)=-2----> a6=a5-2
therefore
Is an arithmetic sequence, the common difference is -2
Part 4) we know that
The y-intercept of the graph is (0,4)
The x-intercept of the graph is (-2,0)
therefore
the function is [tex]y=2x+4[/tex]
because
For x=0 -----> y=2(0)+4 -----> y=4
For y=0 ----> 0=2x+4 --------> x=-2
Part 5) we know that
The formula to calculate the slope between two points is equal to
[tex]m=\frac{y2-y1}{x2-x1}[/tex]
we have
[tex]A(3,5)\ B(2,7)[/tex]
substitute the values
[tex]m=\frac{7-5}{2-3}[/tex]
[tex]m=-2[/tex]
Part 6) we know that
The equation of the line into slope point form is equal to
[tex]y-y1=m(x-x1)[/tex]
we have
[tex]m=4[/tex]
[tex]point(-3,2)[/tex]
substitute the values
[tex]y-2=4(x+3)[/tex]
Convert to slope intercept form
[tex]y=4x+12+2[/tex]
[tex]y=4x+14[/tex]
Part 7) we know that
If two lines are parallel, then their slopes are the same
The equation of the given line is [tex]y=4x-2[/tex]
so
The slope of the given line is [tex]m=4[/tex]
therefore
The line [tex]y=4x+5[/tex] is parallel to the given line
Because the slope is equal to [tex]m=4[/tex]
Part 8) we know that
If a ordered pair is a solution of a system of equations, then the ordered pair must satisfy both equations of the system
Verify each case for (2,3)
case a)
y=2x-1 -----> equation 1
y=x+1 -----> equation 2
Substitute the value of x and the value of y in each equation and then compare the results
Verify equation 1
3=2(2)-1
3=3 -----> is true
Verify equation 2
3=2+1
3=3 -----> is true
therefore
The point (2,3) is a solution of the system of equations case a
case b)
y=2x+1 -----> equation 1
y=x-1 -----> equation 2
Substitute the value of x and the value of y in each equation and then compare the results
Verify equation 1
3=2(2)+1
3=5 -----> is not true
therefore
The point (2,3) is not a solution of the system of equations case b
case c)
y=4x-5 -----> equation 1
y=2x -----> equation 2
Substitute the value of x and the value of y in each equation and then compare the results
Verify equation 1
3=4(2)-5
3=3 -----> is true
Verify equation 2
3=2(2)
3=4 -----> is not true
therefore
The point (2,3) is not a solution of the system of equations case c
