The graph of linear function `g` passes through the points `\left(4,-1\right)`and `\left(-8,\ -4\right),`as shown. What are the slope and y-intercept of the graph `g`
To find the slope of the linear function g, we use the formula:
m = (y2 - y1) / (x2 - x1)
Given points (4, -1) and (-8, -4), we have:
m = (-4 - (-1)) / (-8 - 4)
m = (-3) / (-12)
m = 1/4
Therefore, the slope of the linear function g is 1/4.
To find the y-intercept of the linear function g, we can use the slope-intercept form of the equation:
y = mx + b
Using the point (4, -1) and the slope m = 1/4:
-1 = (1/4)(4) + b
-1 = 1 + b
b = -2
Therefore, the y-intercept of the linear function g is -2.
In conclusion, the slope of the graph g is 1/4 and the y-intercept is -2.
The table shows the total cost of different numbers of square feet of carpet. What is the rate of change of the total cost in dollars with respect to the number of square feet purchased for this linear relationship?
Number of Square Feet Purchased Total Cost (dollars)
120 840
140 980
100 1,120
180 1,200
To find the rate of change of the total cost in dollars with respect to the number of square feet purchased, we first need to determine the slope of the linear relationship.
Using two points from the table, we can calculate the slope using the formula:
m = (y2 - y1) / (x2 - x1)
Let's choose the points (100, 1120) and (180, 1200):
m = (1200 - 1120) / (180 - 100)
m = 80 / 80
m = 1
The slope of the linear relationship is 1.
The rate of change of the total cost in dollars with respect to the number of square feet purchased is equal to the slope of the linear relationship. Therefore, the rate of change is 1 dollar per square foot.