Answer
At values of x less than -2, (x < -2)
y = 1
At velues of x between -2 and 0, (-2 ≤ x < 0)
y = -1.5x - 2
At values of x between 0 and 3, (0 ≤ x < 3)
y = -2
At values of x between 3 and 5, (3 ≤ x < 5)
y = x - 5
At values of x greater than 5, (x > 5)
y = 3x - 15
Explanation
To answer this, we will just break the domains into the x-value regions as obtainable from the graph. Starting from the left hand side
At values of x less than -2, (x < -2)
y = 1
At velues of x between -2 and 0, (-2 ≤ x < 0)
To write the equation of the line at this point, we need to note that
The slope and y-intercept form of the equation of a straight line is given as
y = mx + b
where
y = y-coordinate of a point on the line.
m = slope of the line.
x = x-coordinate of the point on the line whose y-coordinate is y.
b = y-intercept of the line.
For this question,
b = y-intercept = -2
We need to calculate the slope.
For a straight line, the slope of the line can be obtained when the coordinates of two points on the line are known. If the coordinates are (x₁, y₁) and (x₂, y₂), the slope is given as
[tex]Slope=m=\frac{Change\text{ in y}}{Change\text{ in x}}=\frac{y_2-y_1}{x_2-x_1}[/tex]For this question,
(x₁, y₁) and (x₂, y₂) are (-2, 1) and (0, -2)
[tex]\text{Slope = }\frac{-2-1}{0-(-2)}=\frac{-3}{0+2}=\frac{-3}{2}=-1.5[/tex]So, for this region,
y = mx + b
y = -1.5x - 2
At values of x between 0 and 3, (0 ≤ x < 3)
y = -2
At values of x between 3 and 5, (3 ≤ x < 5)
We need to find the equation of the line at this point.
The general form of the equation in point-slope form is
y - y₁ = m (x - x₁)
where
y = y-coordinate of a point on the line.
y₁ = This refers to the y-coordinate of a given point on the line
m = slope of the line.
x = x-coordinate of the point on the line whose y-coordinate is y.
x₁ = x-coordinate of the given point on the line
Point = (x₁, y₁) = (3, -2)
x₁ = 3
y₁ = -2
So, we need to solve for the slope
(x₁, y₁) and (x₂, y₂) are (3, -2) and (5, 0)
[tex]\text{Slope = }\frac{0-(-2)}{5-3}=\frac{0+2}{2}=\frac{2}{2}=1[/tex]y - y₁ = m (x - x₁)
m = 1
x₁ = 3, y₁ = -2
y - y₁ = m (x - x₁)
y - (-2) = 1 (x - 3)
y + 2 = x - 3
y = x - 3 - 2
y = x - 5
At values of x greater than 5, (x > 5)
For these parts, we just do it similarly to the region before this
Point = (x₁, y₁) = (5, 0)
x₁ = 5
y₁ = 0
(x₁, y₁) and (x₂, y₂) are (5, 0) and (6, 3)
[tex]\text{Slope = }\frac{3-0}{6-5}=\frac{3}{1}=3[/tex]y - y₁ = m (x - x₁)
m = 3
x₁ = 5, y₁ = 0
y - y₁ = m (x - x₁)
y - 0 = 3 (x - 5)
y = 3x - 15
Hope this Helps!!!
