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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,
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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.
The error that describes Kelsey's mistake is:
Kelsey graphed the y-intercept on the x-axis.
The correct y-intercept for the equation y = 3x + 1 is (0, 1), which means that the line intersects the y-axis at y = 1 when x = 0. However, in the graph shown, the point (0, 1) is not visible, and instead, the point (−3,0) is labeled as the y-intercept. This point does not lie on the y-axis, and therefore, does not represent the y-intercept of the line. Kelsey must have mistakenly identified the point where the line crosses the x-axis as the y-intercept, which is an error.
The estimated total cost, y, of a child’s toy is partially based on the number of batteries used, x. The cost of one battery is $0.75. The toy itself costs $12. Which equation represents the situation?
If the cost of one battery is $0.75 and the toy itself costs $12, then the total cost of the toy with x batteries can be represented by the equation:
y = 0.75x + 12
The term 0.75x represents the cost of x batteries, and the term 12 represents the cost of the toy itself. The sum of these two terms gives the total cost of the toy with x batteries.
Therefore, the equation that represents the situation is:
y = 0.75x + 12
Four students graphed one linear function each.
Rhett
On a coordinate plane, a line goes through points (negative 4, 0) and (0, 1).
Cameron
On a coordinate plane, a line goes through points (0, 4) and (1, 0).
Ellis
On a coordinate plane, a line goes through points (0, negative 4) and (2, 0).
Braden
On a coordinate plane, a line goes through points (0, 1) and (4, 0).
Which student graphed a linear function with a y-intercept at –4?
Rhett
Cameron
The student who graphed a linear function with a y-intercept of -4 is Ellis.
From the given points, the equation for Ellis' line can be found using the slope-intercept form:
y = mx + b
where m is the slope and b is the y-intercept. Substituting the given points, we get:
-4 = m(0) + b (Since the y-intercept is -4, we can substitute x=0 and y=-4 to solve for b)
b = -4
So the equation for Ellis' line is y = mx - 4. To find the slope, we can use the other given point:
0 = m(2) - 4
m = 2
Therefore, the equation of Ellis' line is y = 2x - 4, which has a y-intercept of -4.
Rhett, Cameron, and Braden do not have a y-intercept of -4.
Eric earns a weekly salary and a commission on each item that he sells. The equation y = 10x + 50 represents the amount of money that Eric earns weekly. Bailey earns a greater weekly salary than Eric but the same commission rate. Which graph could represent the amount of money that Bailey earns weekly, y, based on the number of items sold, x?
A graph titled Bailey's Weekly Earnings has items sold on the x-axis and total earned on the y-axis. A line goes through points (0, 75), (5, 125), (10, 175).
A graph titled Bailey's Weekly Earnings has items sold on the x-axis and total earned on the y-axis. A line goes through points (0, 50), (1, 75), (2, 100).
A graph titled Bailey's Weekly Earnings has items sold on the x-axis and total earned on the y-axis. A line goes through points (0, 75), (1, 100), (2, 125).
A graph titled Bailey's Weekly Earnings has items sold on the x-axis and total earned on the y-axis. A line goes through points (0, 50), (5, 100), (10, 150).
Since Bailey earns a greater weekly salary than Eric but the same commission rate, the expression for Bailey's weekly earnings can be written as y = 10x + b, where b is the amount of Bailey's weekly salary that is greater than 50.
To determine which graph represents this situation, we need to look for the one that passes through the point (0, b + 50) and has the same slope as the line y = 10x + 50. This is because the two lines have the same commission rate of 10 dollars per item sold, but Bailey's line is shifted up by a fixed amount equal to her weekly salary, b.
If we subtract 50 from both sides of the given equation y = 10x + 50, we get:
y - 50 = 10x
This is the slope-intercept form of the equation, with the slope being 10. This means that the line representing Eric's earnings has a slope of 10.
From the given options, the graph that matches these conditions is:
A graph titled Bailey's Weekly Earnings has items sold on the x-axis and total earned on the y-axis. A line goes through points (0, 50), (5, 100), (10, 150).
This graph passes through the point (0, 50), which represents Bailey's weekly salary, and has the same slope of 10 as Eric's line. Therefore, this graph represents the amount of money that Bailey earns weekly based on the number of items sold.
The equation y = negative StartFraction 1 Over 20 EndFraction x + 10 represents the gallons of gasoline that remain in Michelle’s car after she drives x miles. Which graph represents this equation?
A graph titled Driving has total miles traveled on the x-axis and gallons of gasoline on the y-axis. A line goes through points (0, 10) and (40, 8).
A graph titled Driving has total miles traveled on the x-axis and gallons of gasoline on the y-axis. A line goes through points (0, 11) and (80, 8).
A graph titled Driving has total miles traveled on the x-axis and gallons of gasoline on the y-axis. A line goes through points (0, 12) and (125, 6).
A graph titled Driving has total miles traveled on the x-axis and gallons of gasoline on the y-axis. A line goes through points (0, 10) and (150, 0).
The equation y = -1/20 x + 10 represents the gallons of gasoline that remain in Michelle’s car after she drives x miles. To determine which graph represents this equation, we need to find the two points that satisfy the equation and plot them on the coordinate plane.
To find the two points, we can set x = 0 and solve for y to get the y-intercept, and then set y = 0 and solve for x to get the x-intercept:
When x = 0, y = -1/20(0) + 10 = 10
So, the y-intercept is (0, 10).
When y = 0, 0 = -1/20 x + 10
-10 = -1/20 x
x = 200
So, the x-intercept is (200, 0).
Therefore, we have the two points (0, 10) and (200, 0), which we can use to plot the graph.
From the given options, the graph that matches these conditions is:
A graph titled Driving has total miles traveled on the x-axis and gallons of gasoline on the y-axis. A line goes through points (0, 10) and (150, 0).
This graph passes through the points (0, 10) and (200, 0), which represent the y-intercept and x-intercept, respectively, and has the correct slope of -1/20. Therefore, this graph represents the equation y = -1/20x + 10 and the gallons of gasoline that remain in Michelle’s car after she drives x miles.
Which point on the graph represents the y-intercept?
On a coordinate plane, a line goes through points V (negative 4, 4), W (0, 2), Y (2, 1), Z (4, 0).
V
W
Y
Z
To find the y-intercept, we need to locate the point where the line intersects the y-axis, which corresponds to the point where x = 0. From the given points, we can see that the point W has an x-coordinate of 0. Therefore, the point W represents the y-intercept.
So, the answer is:
W