We have here four linear functions, and we have to compare the slopes of them as well as their y-intercept. For this, it is important to remember the general equation for the line in the slope-intercept form:
[tex]y=mx+b[/tex]
Where
• m is the slope of the line
,
• b is the y-intercept of the line (0, b)
The y-intercept is the point where the line passes through the y-axis, and at this point, we have that x = 0.
Finding the slopes and the y-intercept for the four functions
Function 1
We have in this case that the function passes through the points:
• (0, 5), (1, 1)
We can find the equation of this line using the two-point form of the line equation:
[tex]y-y_1=\frac{y_2-y_1}{x_2-x_1}(x-x_1)[/tex]
Using the points above, we can label them as follows:
• (0, 5) ---> x1 = 0, y1 = 5
,
• (1, 1) ---> x2 = 1, y2 = 1
Then we have:
[tex]\begin{gathered} y-5_{}=\frac{1_{}-5}{1-0_{}}(x-0_{}) \\ y-5=-\frac{4}{1}x \\ y-5=-4x \\ y=-4x+5 \end{gathered}[/tex]
Therefore, the slope for this line is m = -4, and the y-intercept is (0, 5).
Function 2
For Function 2 we need to do the same procedure as before. We have to select two points of the function to find its equation:
We can select the following points: (0, 1), (1, 6).
Then we can proceed as we did in the previous part:
• (0, 1) ---> x1 = 0, y1 = 1
,
• (1, 6) ---> x2 = 1, y2 = 6
[tex]\begin{gathered} y-1_{}=\frac{6_{}-1_{}}{1-0_{}}(x-0) \\ y-1=5x \\ y=5x+1 \end{gathered}[/tex]
Therefore, the slope in Function 2 is 5, m = 5, and the y-intercept is (0, 1).
Function 3
We can see that Function 3 has already its line equation in slope-intercept form:
[tex]y=-x-2[/tex]
Then the slope for Function 3 is m = -1, and its y-intercept is (0, -2).
Function
From the question, we have:
