a To graph a line, we have to identify at least two points that fall on the line. Examining the equations, we see that both equations are written in slope-intercept form.
y=mx+b
In this equation, m is the line's slope and b is the y-intercept. We can identify both m and b by examining the equations.
|c|c|c|
[-0.8em]
Function & m & b [0.3em]
[-0.6em]
y=2x+ & 2 & [0.5em]
[-0.8em]
y=-1/2x+6 & -1/2 & 6 [0.8em]
Let's plot the y-intercepts of our functions
To find a second point on the lines we will use their slope. When we have two points their lines can be drawn.
Examining the diagram, the intercepts of both functions can be identified.
|c|c|c|
[-0.8em]
Function & x-intercept & y-intercept [0.3em]
[-0.6em]
y=2x+0 & (0,0) & (0,0) [0.5em]
[-0.8em]
y=-1/2x+6 & (12,0) & (0,6) [0.8em]
b Below, we have shaded the region between the lines and the x-axis in the diagram.
c The domain refers to the allowed x-values. From the diagram, we see that the region goes from x=0 to x=12 including the endpoints.
With this information, we can write the domain.
0 ≤ x ≤ 12
The range is all of the y-values that the region can take on. From the diagram we notice that the range goes from y=0 to the graph's point of intersection.
To find the y-coordinate of the intersection we must first find its x-coordinate. We can determine this by equating the functions and solving for x.
Now we can find the y-coordinates by substituting x= 2.4 into either equation and simplifying.
y=2( 2.4) ⇔ y = 4.8
Now we can identify the range.
0 ≤ y ≤ 4.8
d The region is a triangle which means we can find its area by using the following formula.
A=1/2bh
If we let the domain be the triangle's base, then the range must be the triangle's height.
