Determine an expression of the linear functions f whose graphs are the following ones. If no function exists, determine the equation of the straight line.
In order to determine the equation of the function, we need to find two points that the line passes through. The easiest points to work with are the y -intercept and the x -intercept:
- A=\left(0{,}2\right)
- B=\left(4{,}0\right)
The slope of the line is:
m=\dfrac{y_B-y_A}{x_B-x_A}
Here, we have:
m=\dfrac{0-2}{4-0}= -\dfrac{1}{2}
Furthermore, for any point B on the line, we have:
f\left(x\right)-y_B=m\left(x-x_B\right)
Therefore:
f\left(x\right) - 0 = -\dfrac{1}{2}\left(x-4\right)
f\left(x\right) = -\dfrac{1}{2}x + 2
f\left(x\right)=-0.5x+2
The y -intercept and the x -intercept are:
- A=\left(0{,}1\right)
- B=\left(1{,}0\right)
The slope of the line is:
m=\dfrac{y_B-y_A}{x_B-x_A}
Here, we have:
m=\dfrac{0-1}{1-0}= -1
Furthermore, for any point B on the line, we have:
f\left(x\right)-y_B=m\left(x-x_B\right)
Therefore:
f\left(x\right) - 0 = -{1}\left(x-1\right)
f\left(x\right) = -x + 1
f\left(x\right) = -x + 1
The y -intercept and the x -intercept are:
- A=\left(0{,}1\right)
- B=\left(-1{,}0\right)
The slope of the line is:
m=\dfrac{y_B-y_A}{x_B-x_A}
Here, we have:
m=\dfrac{0-1}{-1-0}= 1
Furthermore, for any point B on the line, we have:
f\left(x\right)-y_B=m\left(x-x_B\right)
Therefore:
f\left(x\right) - 0 = {1}\left(x-\left(-1\right)\right)
f\left(x\right) = x + 1
f\left(x\right) = x + 1
The graph is a straight line parallel to the x-axis at a distance of two units above the x-axis. Therefore an expression is:
f\left(x\right)=2
f\left(x\right)=2
The graph is a straight line parallel to the y-axis at a distance of one unit left of the y-axis. Therefore an equation is:
x=-1
x=-1
The y -intercept and the x -intercept are:
- A=\left(0{,}2\right)
- B=\left(1{,}0\right)
The slope of the line is:
m=\dfrac{y_B-y_A}{x_B-x_A}
Here, we have:
m=\dfrac{2-0}{0-1}= -2
Furthermore, for any point B on the line, we have:
f\left(x\right)-y_B=m\left(x-x_B\right)
Therefore:
f\left(x\right) - 0 = -{2}\left(x-1\right)
f\left(x\right) = -2x + 2
f\left(x\right) = -2x + 2
The y -intercept and the x -intercept are:
- A=\left(0,-1\right)
- B=\left(\dfrac{1}{2},0\right)
The slope of the line is:
m=\dfrac{y_B-y_A}{x_B-x_A}
Here, we have:
m=\dfrac{0-\left(-1\right)}{1/2-0}= 2
Furthermore, for any point B on the line, we have:
f\left(x\right)-y_B=m\left(x-x_B\right)
Therefore:
f\left(x\right) - 0 = 2\left(x-\dfrac{1}{2}\right)
f\left(x\right) = 2x - 1
f\left(x\right) = 2x - 1