Determine if 7x2+y=5 is a linear or nonlinear function.
y=−25x+3. Enter 1 for yes. Enter 2 for no.
Is the slope of the line in the equation y = -25x + 3 negative?
1. Yes
2. No
Determine the initial value and the rate of change in the following description.
Last year, Joshua’s tomato plant reached a maximum height of 2 feet. He is growing another tomato plant this year, and he is tracking its growth. The plant’s height today is 10 inches, and it grows 1.7 inches each day.
Initial value: 2 feet = 24 inches (since the maximum height reached by Joshua's tomato plant last year was 2 feet)
Rate of change: 1.7 inches per day (since the plant grows 1.7 inches each day)
x y
2 22
8 25
Determine the rate of change and the initial value of the linear function given here as a table of values.
To determine the rate of change and the initial value of the linear function, we need to find the slope and y-intercept.
First, let's find the slope using the two points given (2, 22) and (8, 25):
slope = change in y / change in x
= (25 - 22) / (8 - 2)
= 3 / 6
= 1/2
So the slope is 1/2.
Next, we can use one of the points and the slope to find the y-intercept. Let's use the point (2, 22):
y = mx + b (where m is the slope and b is the y-intercept)
22 = (1/2)(2) + b
22 = 1 + b
b = 21
So the y-intercept is 21.
Therefore, the linear function is:
y = (1/2)x + 21
The rate of change is 1/2 and the initial value is 21.
(-3,4) (0,3) (3,2) (6,1)
Determine the initial value and the rate of change of the linear function as given in the graph. Round the answer to three decimal places as needed.
To determine the initial value and rate of change of the linear function, we need to first find the slope, which is the rate of change.
Using the two points (3, 2) and (6, 1):
slope = change in y / change in x
= (1 - 2) / (6 - 3)
= -1/3
So the slope is -1/3.
Next, we can use any of the points and the slope to find the y-intercept. Let's use point (0, 3):
y = mx + b (where m is the slope and b is the y-intercept)
3 = (-1/3)(0) + b
3 = b
So the y-intercept is 3.
Therefore, the linear function is:
y = (-1/3)x + 3
The rate of change is -1/3 and the initial value is 3.
A pan with four eggs weighs 18 ounces. A pan with eight eggs weighs 24 ounces. What is the weight of the pan?
We can start the problem by using algebra. Let "p" be the weight of the pan, in ounces.
According to the problem, a pan with four eggs weighs 18 ounces. This means that the weight of the eggs is 18 - p ounces. Similarly, a pan with eight eggs weighs 24 ounces, which means that the weight of the eggs is 24 - p ounces.
We can set up an equation based on the above information:
24 - p = 2(18 - p)
In words, this equation says that the weight of the eight-egg pan (24 - p) is twice the weight of the four-egg pan (2 * (18 - p)).
Now we can solve for "p":
24 - p = 36 - 2p
p = 12
Therefore, the weight of the pan is 12 ounces.
Manuel got a new job and was given a hiring bonus along with his monthly salary. His earnings after x months are modeled by the function f(x)=1,500x+300. How much did he receive as a hiring bonus?
In the given function, the constant term (300) represents the hiring bonus, and the coefficient of x (1500) represents Manuel's monthly salary.
Therefore, Manuel received a hiring bonus of $300.
y=mx+b defines what type of function?
The equation y = mx + b represents a linear function, where:
- m is the slope of the line
- b is the y-intercept of the line.
x y
6 16
15 43
Write the equation of the linear function that models the relationship shown in the table. Enter your answer in y=mx+b form.
To write the equation of the linear function that models the relationship shown in the table, we need to find the slope of the line passing through the two points given.
Using the formula for slope:
slope = (y2 - y1) / (x2 - x1)
where (x1, y1) = (6, 16) and (x2, y2) = (15, 43), we get:
slope = (43 - 16) / (15 - 6)
= 27 / 9
= 3
Now that we have the slope (m = 3), we can use the point-slope form of a linear equation to write the equation:
y - y1 = m(x - x1)
where (x1, y1) is either of the two points. Let's use (6, 16):
y - 16 = 3(x - 6)
Simplifying:
y - 16 = 3x - 18
y = 3x - 2
Therefore, the equation of the linear function that models the relationship shown in the table is y = 3x - 2.
Last year, Alayah ordered 4 cubic yards of soil and spent $220. This year, she orders 7 cubic yards of soil and spends $340. Write the equation of the linear function that models the relationship between the number of cubic yards of soil ordered and the total cost. Enter your answer in y=mx+b form.
To write the equation of the linear function that models the relationship between the number of cubic yards of soil ordered and the total cost, we need to find the slope and y-intercept of the line passing through the two points given.
Using the formula for slope:
slope = (y2 - y1) / (x2 - x1)
where (x1, y1) = (4, 220) and (x2, y2) = (7, 340), we get:
slope = (340 - 220) / (7 - 4)
= 120 / 3
= 40
Now that we have the slope (m = 40), we can use the point-slope form of a linear equation to write the equation:
y - y1 = m(x - x1)
where (x1, y1) is either of the two points. Let's use (4, 220):
y - 220 = 40(x - 4)
Simplifying:
y - 220 = 40x - 160
y = 40x + 60
Therefore, the equation of the linear function that models the relationship between the number of cubic yards of soil ordered and the total cost is y = 40x + 60.
Compare Linear Function 1 with Linear Function 2. Which one has the greater rate of change? Choose 1 for Linear Function 1; choose 2 for Linear Function 2.
Linear Function 1: y=x+8
Linear Function 2:
A coordinate plane shows the x-axis ranging from negative 2 to 6 in increments of 1 and the y-axis ranging from negative 2 to 12 in increments of 1. A line with arrows at both ends joins two plotted points. The coordinates of the plotted points are as follows: left parenthesis 2 comma 6 right parenthesis and left parenthesis 4 comma 10 right parenthesis.
Linear Function 1: y = x + 8
Linear Function 2: The plotted points are (2, 6) and (4, 10). To find the equation, we can find the slope first:
slope = change in y / change in x
= (10 - 6) / (4 - 2)
= 2
Now we can use one of the points and the slope to find the y-intercept:
y = mx + b
6 = 2(2) + b
6 = 4 + b
b = 2
So the equation of the line passing through the two points is y = 2x + 2.
Comparing the two functions, we can see that the rate of change (which is the same as the slope) of Linear Function 2 is greater than the rate of change of Linear Function 1.
Therefore, the answer is 2 for Linear Function 2 having the greater rate of change.
Compare Linear Function 1 with Linear Function 2. Which one has the greater rate of change? Choose 1 for Linear Function 1; choose 2 for Linear Function 2.
Linear Function 1: y=x+8
Linear Function 2:
(4,10) (2,6)
Linear Function 1: y = x + 8
Linear Function 2: The given points are (4, 10) and (2, 6). To find the slope, we use the formula:
slope = change in y / change in x
= (10 - 6) / (4 - 2)
= 2
Now we can compare the slopes of the two functions. The slope of Linear Function 1 is 1, which is less than the slope of Linear Function 2, which is 2.
Therefore, the answer is 2 for Linear Function 2 having the greater rate of change.
Day of the Week Number of Children in the Car
Sunday 2
Monday 4
Tuesday 4
Wednesday 3
Thursday 4
Friday 3
Saturday 0
Describe the meaning of the word function in math. Then consider the table. Why does this table show a function?
In mathematics, a function is a rule that assigns to each input exactly one output.
This table shows a function because for each day of the week, there is a unique number of children in the car. Each input (day of the week) has exactly one output (number of children in the car). For example, on Sunday, there were 2 children in the car, and on Monday, there were 4 children in the car. No day has more than one output for the number of children in the car, which satisfies the definition of a function.