A graph has time (hours) on the x-axis and total charge (dollars) on the y-axis. Points are at (0, 8), (2, 27), (4, 46), (6, 65).
For the new year, the instructor is thinking of changing his rates. The equation of the possible new rates is represented in the graph. What is the y-intercept of the instructor's new rates?
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To find the y-intercept of the instructor's new rates, we need to look for the point where the graph crosses the y-axis, which corresponds to the value of total charge when the time is 0. From the given points, we can observe that the total charge increases by 19 dollars for every 2-hour increase in time. Therefore, the rate of change (or slope) of the graph is 19/2 = 9.5 dollars per hour.
To find the y-intercept, we can use the slope-intercept form of the equation of a line, which is y = mx + b, where m is the slope and b is the y-intercept. Substituting the slope and the coordinates of one of the points, say (0, 8), we get:
8 = 9.5(0) + b
b = 8
Therefore, the y-intercept of the instructor's new rates is 8 dollars.
he instructor had saved $3,500 and rents an apartment for $275 monthly. He believes the point (6, 1850) would be on the equation of the line. Is he correct? Explain how to check if a point is on a line.
To check if a point is on a line, we need to substitute the coordinates of the point into the equation of the line and see if the equation is true. For a linear equation in slope-intercept form (y = mx + b), we can calculate y by substituting the x-value of the point into the equation and checking if it matches the y-value of the point.
In this case, we do not have an equation for the line, but we can find it using the given information. Since the instructor rents an apartment for $275 monthly, we can assume that his monthly expenses are $275, and the remaining amount of his savings after each month is $3,500 - $275x, where x is the number of months. If he believes that the point (6, 1850) would be on the equation of the line, it means that he thinks his savings after 6 months would be $1850.
We can check if this is true by substituting x = 6 into the expression for his savings after x months:
savings = $3,500 - $275x
savings = $3,500 - $275(6)
savings = $3,500 - $1,650
savings = $1,850
We see that the savings after 6 months is indeed $1,850, which matches the y-value of the given point. Therefore, the instructor is correct in thinking that the point (6, 1850) would be on the equation of the line. We can use this point and any other point on the line to find the equation in slope-intercept form and check that the point is indeed on the line.
Samantha’s parents give her $10 per week. For each chore that she does not complete, she must repay her parents $1. Which graph models Samantha’s allowance?
A graph titled Samantha's Allowance has chores not completed on the x-axis and amount earned on the y-axis. A line goes through points (0, 10) and (1, 9).
A graph titled Samantha's Allowance has chores not completed on the x-axis and amount earned on the y-axis. A line goes through points (0, 10) and (2, 12).
A graph titled Samantha's Allowance has chores not completed on the x-axis and amount earned on the y-axis. A line goes through points (1, 6) and (1.5, 16).
A graph titled Samantha's Allowance has chores not completed on the x-axis and amount earned on the y-axis. A line goes through points (0, 10) and (1, 0).
The correct graph that models Samantha's allowance is the fourth option:
A graph titled Samantha's Allowance has chores not completed on the x-axis and amount earned on the y-axis. A line goes through points (0, 10) and (1, 0).
This graph represents the scenario where Samantha earns $10 per week initially, but for each chore that she does not complete, she loses $1. Therefore, if she completes all her chores, she keeps the full $10, but if she doesn't complete any chore, she loses the full $10 and ends up with 0 earnings.
The line passing through the points (0, 10) and (1, 0) represents this linear relationship between the number of chores not completed and the amount earned. When no chores are completed (x = 1), Samantha doesn't earn anything (y = 0), and for each additional chore not completed, she loses $1 (the slope of the line is -1). Therefore, this graph models Samantha's allowance.
What is the slope, m, and y-intercept for the line that is plotted on the grid below?
On a coordinate plane, a line goes through points (0, 4) and (negative 2, 0).
m = One-half, (0, –2)
m = One-half, (0, 4)
m = 2, (0, –2)
m = 2, (0, 4)
To find the slope, we use the formula:
m = (y2 - y1) / (x2 - x1)
where (x1, y1) and (x2, y2) are the coordinates of the two points on the line.
Using the given points (0, 4) and (-2, 0), we have:
m = (0 - 4) / (-2 - 0) = -4 / (-2) = 2
Therefore, the slope of the line is 2.
To find the y-intercept, we can use the slope-intercept form of the equation of a line, which is:
y = mx + b
where m is the slope and b is the y-intercept.
Substituting the slope and the coordinates of one of the points, say (0, 4), we get:
4 = 2(0) + b
b = 4
Therefore, the y-intercept of the line is 4.
Thus, the answer is: m = 2, (0, 4)
A sand dune stands 5 feet above sea level. The hill is eroding at a rate of 1 foot per 20 years. Let y represent the height of the sand dune after x years. Which equation represents the situation?
y = negative StartFraction 1 Over 20 EndFraction x minus 5
y = negative StartFraction 1 Over 20 EndFraction x + 5
y = negative 20 x minus 5
y = negative 20 x + 5
Since the sand dune is 5 feet above sea level and eroding at a rate of 1 foot per 20 years, the height of the sand dune after x years can be represented by:
y = -1/20 x + 5
The negative sign in front of 1/20 indicates that the height of the sand dune is decreasing over time. The constant term, 5, represents the initial height of the sand dune above sea level.
Therefore, the equation that represents the situation is:
y = negative StartFraction 1 Over 20 EndFraction x + 5
Kelsey graphed the equation y = 3x + 1 as shown below.
On a coordinate plane, a line goes through points (negative 3, 0) and (0, 3).
Which describes Kelsey’s error?
Kelsey graphed the y-intercept on the x-axis.
Kelsey graphed –3 instead of 3 as the y-intercept.
Kelsey graphed the slope as up 1 right 1 instead of up 1 left 1.
Kelsey graphed the slope as the y-intercept and the y-intercept as the slope.