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Determine if the relation \( y = 4x - 1 \) is a function:
- The equation \( y = 4x - 1 \) is a linear equation, and when graphed, it produces a straight line.
- The vertical line test states that if a vertical line intersects the graph of a relation at most once for every value of \( x \), then the relation is a function. Since a straight line passes this test, the relation is indeed a function.
- Answer: 1
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Which of the following tables represents a function that is proportional?
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Function 1:
- \( x \) | \( y \)
- ---- | ----
- 1 | 2
- 2 | 4
- 3 | 6
The ratio \( \frac{y}{x} \) for Function 1 is \( \frac{2}{1} = 2 \), \( \frac{4}{2} = 2 \), \( \frac{6}{3} = 2 \). All ratios are equal, indicating a proportional relationship.
-
Function 2:
- \( x \) | \( y \)
- ---- | ----
- 1 | 2
- 2 | 3
- 3 | 4
The ratios are \( \frac{2}{1} = 2 \), \( \frac{3}{2} = 1.5 \), \( \frac{4}{3} \approx 1.33 \). The ratios differ, indicating it is not proportional.
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Answer: Function 1 is proportional.
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Determine if the given graph is the graph of the function \( y = -25x + 3 \):
- The description indicates that the line is downward slanting, which is consistent with a negative slope. However, to confirm if it matches the specific equation given, we'd need to verify the y-intercept and slope.
- The slope (-25) is very steep, and the y-intercept is 3. If the points (0, 3) and (5, 1) are accurately plotted and the slope calculated from these is checked, the line’s slope is actually \( \frac{1-3}{5-0} = -\frac{2}{5} \) which does not equal -25.
- Answer: 2 (no)
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Determine the initial value and the rate of change based on the description of Joshua's tomato plant:
- The initial height today is 10 inches (or 0.833 feet, since there are 12 inches in a foot), and the plant grows at a rate of 1.7 inches per day.
- This situation can be described by the equation \( h(t) = h_0 + mt \) where \( h_0 \) is the initial height, and \( m \) is the rate of growth.
- Here, \( h_0 = 10 \) inches, which means \( b = 10 \), and the growth rate \( m = 1.7 \).
- Answer: \( b = 10 \), \( m = 1.7 \)
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Determine the rate of change and the initial value of the linear function based on the table:
- Given points: \( (2, 22) \) and \( (8, 25) \).
- To find the rate of change (slope \( m \)): \[ m = \frac{y_2 - y_1}{x_2 - x_1} = \frac{25 - 22}{8 - 2} = \frac{3}{6} = 0.5 \]
- To find the initial value (y-intercept \( b \)), we can use the slope-intercept form \( y = mx + b \). Using one of the points (2, 22): \[ 22 = 0.5(2) + b \implies 22 = 1 + b \implies b = 21 \]
- Answer: \( m = 0.5 \), \( b = 21 \)
Determine if the relation y=4x−1
is a function by first graphing the relation, then applying the vertical line test. Enter 1 if the relation is a function. Enter 2 if the relation is not a function.(1 point)
Which of the following tables represents a function that is proportional?
Function 1:
x
y
1 2
2 4
3 6
Function 2:
x
y
1 2
2 3
3 4
(1 point)
Function
blank is proportional
Use the image to answer the question.
An illustration shows a coordinate plane with four quadrants. The x-axis and y-axis both range from negative 6 to 6 in one unit increments. A downward slanting line is graphed on the coordinate plane. The line passes through points plotted at left parenthesis 0 comma 3 right parenthesis, and left parenthesis 5 comma 1 right parenthesis.
Determine if the given graph is the graph of the function y=−25x+3
. Enter 1 for yes. Enter 2 for no.
(1 point)
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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.
(1 point)
The initial value is b=
, and the rate of change is m=
.
Use the table to answer the question.
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.
(1 point)
The rate of change is m=
, and the initial value is b=
.
7 answers
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.
(1 point)
The initial value is
, and the rate of change is
.
Functions Unit Test
7 of 157 of 15 Items
Question
Use the image to answer the question.
A coordinate plane shows quadrant 4 with the x-axis, labeled as minutes, ranging from 0 to 6 in increments of 1 and y-axis, labeled as elevation in feet, ranging from 0 to negative 110 in decrements of 10. A ray with arrow at the bottom end points downward from the origin. The coordinates that it passes through are left parenthesis 0 comma 0 right parenthesis, left parenthesis 3 comma negative 60 right parenthesis, and left parenthesis 5 comma negative 100 right parenthesis.
The graph shows the elevation of a deep-sea diver in relation to minutes passed. Using any two points on the graph, what is the rate of change in the diver’s elevation?
(1 point)
feet per minute
A pan with four eggs weighs 18 ounces. A pan with eight eggs weighs 24 ounces. What is the weight of the pan?
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?(1 point)
The equation y=mx+b
defines what type of function?(1 point)
Responses
a linear function
a linear function
an initial value function
an initial value function
a slope function
a slope function
a nonlinear function
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Determine the initial value and the rate of change of the linear function from the graph:
- The points given are:
- (-3, 4)
- (0, 3)
- (3, 2)
- (6, 1)
To find the rate of change (slope \( m \)): \[ m = \frac{y_2 - y_1}{x_2 - x_1} \] Using the points (0, 3) and (6, 1): \[ m = \frac{1 - 3}{6 - 0} = \frac{-2}{6} = -\frac{1}{3} \approx -0.333 \]
Initial value \( b \) is the value of \( y \) when \( x = 0 \): From the point (0, 3), the initial value \( b = 3 \).
- Answer: The initial value is 3, and the rate of change is -0.333.
- The points given are:
-
Use the graph to find the rate of change in the diver’s elevation:
- The points given are:
- (0, 0)
- (3, -60)
- (5, -100)
Using the points (0, 0) and (3, -60): \[ m = \frac{-60 - 0}{3 - 0} = \frac{-60}{3} = -20 \] Hence, the rate of change is \(-20\) feet per minute.
- Answer: -20 feet per minute.
- The points given are:
-
Determine the weight of the pan with four eggs and eight eggs:
- Let \( w \) be the weight of the pan and \( e \) be the weight of one egg.
- From the problem:
- For four eggs: \( w + 4e = 18 \)
- For eight eggs: \( w + 8e = 24 \)
We can set up a system of equations: \[ w + 4e = 18 \quad \text{(1)} \] \[ w + 8e = 24 \quad \text{(2)} \]
Subtract equation (1) from equation (2): \[ (w + 8e) - (w + 4e) = 24 - 18 \ 4e = 6 \ e = \frac{6}{4} = 1.5 \quad \text{(weight of one egg)} \]
Substitute \( e = 1.5 \) in equation (1): \[ w + 4(1.5) = 18 \ w + 6 = 18 \ w = 12 \]
- Answer: The weight of the pan is 12 ounces.
- Determine how much Manuel received as a hiring bonus:
-
The function given is \( f(x) = 1,500x + 300 \), where \( 1,500 \) represents the monthly salary, and \( 300 \) represents the hiring bonus.
-
Therefore, the hiring bonus is 300.
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Answer: The hiring bonus is 300.
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- What type of function does the equation \( y = mx + b \) define?
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The equation \( y = mx + b \) is the slope-intercept form of a linear equation.
-
Hence, it defines:
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Answer: a linear function.
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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.
(1 point)
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.(1 point)
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.
(1 point)
Linear Function
has the greater rate of change.
A function has an initial value of 2 and a slope equal to −1
. Choose the sketch of a graph that exhibits the qualitative features of the function.(1 point)
Responses
A coordinate plane shows the x and y axes ranging from negative 8 to 8 in increments of 1. An upward line with arrows at both ends passes through the first, second, and third quadrants. The line intersects the x-axis at left parenthesis negative 1 comma 0 right parenthesis, and the y-axis at left parenthesis 0 comma 2 right parenthesis.
Image with alt text: A coordinate plane shows the x and y axes ranging from negative 8 to 8 in increments of 1. An upward line with arrows at both ends passes through the first, second, and third quadrants. The line intersects the x-axis at left parenthesis negative 1 comma 0 right parenthesis, and the y-axis at left parenthesis 0 comma 2 right parenthesis.
A coordinate plane shows the x and y axes ranging from negative 8 to 8 in increments of 1. An upward line with arrows at both ends passes through the first, second, and third quadrants. The line intersects the x-axis at left parenthesis negative 2 comma 0 right parenthesis, and the y-axis at left parenthesis 0 comma 2 right parenthesis.
Image with alt text: A coordinate plane shows the x and y axes ranging from negative 8 to 8 in increments of 1. An upward line with arrows at both ends passes through the first, second, and third quadrants. The line intersects the x-axis at left parenthesis negative 2 comma 0 right parenthesis, and the y-axis at left parenthesis 0 comma 2 right parenthesis.
A coordinate plane shows the x and y axes ranging from negative 8 to 8 in increments of 1. A downward line with arrows at both ends passes through the second, first, and fourth quadrants. The line intersects the x-axis at left parenthesis 2 comma 0 right parenthesis, and the y-axis at left parenthesis 0 comma 2 right parenthesis.
Image with alt text: A coordinate plane shows the x and y axes ranging from negative 8 to 8 in increments of 1. A downward line with arrows at both ends passes through the second, first, and fourth quadrants. The line intersects the x-axis at left parenthesis 2 comma 0 right parenthesis, and the y-axis at left parenthesis 0 comma 2 right parenthesis.
A coordinate plane shows the x and y axes ranging from negative 8 to 8 in increments of 1. An upward line with arrows at both ends passes through the first, fourth, and third quadrants. The line passes through left parenthesis 2 comma 3 right parenthesis, and left parenthesis negative 2 comma negative 5 right parenthesis.
Image with alt text: A coordinate plane shows the x and y axes ranging from negative 8 to 8 in increments of 1. An upward line with arrows at both ends passes through the first, fourth, and third quadrants. The line passes through left parenthesis 2 comma 3 right parenthesis, and left parenthesis negative 2 comma negative 5 right parenthesis.
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Let's work through each question one by one.
1. Write the equation of the linear function from the table:
Given points:
- \( (6, 16) \)
- \( (15, 43) \)
Step 1: Calculate the slope \( m \)
\[ m = \frac{y_2 - y_1}{x_2 - x_1} \]
Using points \( (6, 16) \) and \( (15, 43) \):
\[ m = \frac{43 - 16}{15 - 6} = \frac{27}{9} = 3 \]
Step 2: Write the equation in \( y = mx + b \) form.
Now we can use one of the points to solve for \( b \). Let's use the point \( (6, 16) \):
\[ y = mx + b \implies 16 = 3(6) + b \]
\[ 16 = 18 + b \implies b = 16 - 18 = -2 \]
Thus, the equation is:
\[ y = 3x - 2 \]
2. Write the equation of the linear function for soil cost:
Given:
- Last year: \( 4 \) cubic yards and \( 220 \) dollars.
- This year: \( 7 \) cubic yards and \( 340 \) dollars.
Step 1: Calculate the slope \( m \)
\[ m = \frac{340 - 220}{7 - 4} = \frac{120}{3} = 40 \]
Step 2: Write the equation in \( y = mx + b \) form.
Using one of the points, let's use \( (4, 220) \):
\[ y = mx + b \implies 220 = 40(4) + b \]
\[ 220 = 160 + b \implies b = 220 - 160 = 60 \]
Thus, the equation is:
\[ y = 40x + 60 \]
3. Compare Linear Function 1 with Linear Function 2 and find the greater rate of change:
- Linear Function 1: \( y = x + 8 \) has a slope (\( m = 1 \)).
- Linear Function 2: The points \( (2, 6) \) and \( (4, 10) \) can be used to find the slope:
\[ m = \frac{10 - 6}{4 - 2} = \frac{4}{2} = 2 \]
Conclusion
Linear Function 2 has the greater rate of change.
4. Identify the graph that represents a function with an initial value of 2 and slope of -1:
The function with an initial value (y-intercept \( b = 2 \)) and a slope of -1 can be represented by the equation:
\[ y = -x + 2 \]
In this case, the graph would be a line that intersects the y-axis at \( (0, 2) \) and has a negative slope.
From the descriptions given, look for a downward line that intersects:
- Y-axis at \( (0, 2) \)
Final Answer for the Graph:
If any of the options list a graph that has the y-intercept at \( (0, 2) \) and slopes downwards, that would be your correct choice.
If we don't actually see the images to select from, please verify based on the slope and y-intercept attributes as described above.
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?
Meaning of Function in Mathematics
In mathematics, a function is a specific relation between a set of inputs and a set of outputs. Each input corresponds to exactly one output. Functions can be described in various forms including equations, tables, and graphs. The key characteristics of a function are:
- Unique Output: For each input (often represented as \(x\)), there is exactly one output (often represented as \(y\)). This means that no input can point to more than one output.
- Well-defined Relation: The relationship between the input and output must be consistent throughout the function.
For example, in a function defined by the equation \(y = 2x\), if you input 3 (as \(x\)), you will always get 6 (as \(y\)), and there is no other output for that input.
Why the Table Shows a Function
Looking at the provided table:
| Day of the Week | Number of Children in the Car | |------------------|-------------------------------| | Sunday | 2 | | Monday | 4 | | Tuesday | 4 | | Wednesday | 3 | | Thursday | 4 | | Friday | 3 | | Saturday | 0 |
Analysis of the Table
- Inputs: The inputs are the days of the week (Sunday, Monday, Tuesday, etc.).
- Outputs: The outputs are the number of children in the car for each respective day.
To determine if this table represents a function:
- Each day of the week corresponds to a single number of children.
- There are no repeated days with different counts; for each day listed, there is only one number of children.
Although the same number of children appears on different days (for example, 4 children on Monday, Tuesday, and Thursday), this does not violate the definition of a function, as the input (day of the week) still yields one unique output (number of children).
Conclusion
This table represents a function because:
- Each day (input) maps to exactly one value (output, the number of children).
- Thus, the relationship between "day of the week" and "number of children in the car" is consistent and adheres to the definition of a function in mathematics.
1 answer