Part 1) option a. ![y=(x+1)^{2}](/tpl/images/0456/1717/6a316.png)
Part 2) option c. ![y(x)=10x+1](/tpl/images/0456/1717/82a7c.png)
Part 3) option a. Yes , d=-2
Part 4) option b. ![y=2x+4](/tpl/images/0456/1717/231b2.png)
Part 5) option b. ![m=-2](/tpl/images/0456/1717/3a130.png)
Part 6) option c. ![y=4x+14](/tpl/images/0456/1717/3c22e.png)
Part 7) option c. ![y=4x+5](/tpl/images/0456/1717/35378.png)
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
![y=(x+1)^{2}](/tpl/images/0456/1717/6a316.png)
For x=1, y=4
![4=(1+1)^{2}](/tpl/images/0456/1717/148cf.png)
----> is true
For x=2, y=9
![9=(2+1)^{2}](/tpl/images/0456/1717/65397.png)
----> is true
For x=3, y=16
![16=(3+1)^{2}](/tpl/images/0456/1717/e41a8.png)
----> is true
therefore
The function pass through the three points
case b) we have
![y=(x+3)^{2}](/tpl/images/0456/1717/6d548.png)
For x=1, y=4
![4=(1+3)^{2}](/tpl/images/0456/1717/8fc84.png)
----> is not true
therefore
The function not pass through the three points
case c) we have
![y=7x-5](/tpl/images/0456/1717/fdc7a.png)
For x=1, y=4
![4=7(1)-5](/tpl/images/0456/1717/506da.png)
----> 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
![y(x)=10x+1](/tpl/images/0456/1717/82a7c.png)
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 ![y=2x+4](/tpl/images/0456/1717/231b2.png)
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
![m=\frac{y2-y1}{x2-x1}](/tpl/images/0456/1717/baca9.png)
we have
![A(3,5)\ B(2,7)](/tpl/images/0456/1717/10441.png)
substitute the values
![m=\frac{7-5}{2-3}](/tpl/images/0456/1717/268a2.png)
![m=-2](/tpl/images/0456/1717/3a130.png)
Part 6) we know that
The equation of the line into slope point form is equal to
![y-y1=m(x-x1)](/tpl/images/0456/1717/11c10.png)
we have
![m=4](/tpl/images/0456/1717/b848a.png)
![point(-3,2)](/tpl/images/0456/1717/85b3c.png)
substitute the values
![y-2=4(x+3)](/tpl/images/0456/1717/f5465.png)
Convert to slope intercept form
![y=4x+12+2](/tpl/images/0456/1717/b06a0.png)
![y=4x+14](/tpl/images/0456/1717/3c22e.png)
Part 7) we know that
If two lines are parallel, then their slopes are the same
The equation of the given line is ![y=4x-2](/tpl/images/0456/1717/8ce54.png)
so
The slope of the given line is ![m=4](/tpl/images/0456/1717/b848a.png)
therefore
The line
is parallel to the given line
Because the slope is equal to ![m=4](/tpl/images/0456/1717/b848a.png)
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