Which of the following tables represents a function that is proportional?

Question

Function 1:

x
y

1 2
2 4
3 6
Function 2:

x
y

1 2
2 3
3 4
(1 point)

Answer 1

A function is proportional if there exists a constant \( k \) such that \( y = kx \). This means that for a proportional relationship, the ratio \( \frac{y}{x} \) must be the same for all pairs of \( (x, y) \).

Let's analyze both functions:

Function 1:

\[ \begin{array}{|c|c|} \hline x & y \ \hline 1 & 2 \ 2 & 4 \ 3 & 6 \ \hline \end{array} \]

Calculating the ratios \( \frac{y}{x} \):

• For \( (1, 2) \): \( \frac{2}{1} = 2 \)
• For \( (2, 4) \): \( \frac{4}{2} = 2 \)
• For \( (3, 6) \): \( \frac{6}{3} = 2 \)

Since the ratio \( \frac{y}{x} \) is the same (2) for all pairs, Function 1 is proportional.

Function 2:

\[ \begin{array}{|c|c|} \hline x & y \ \hline 1 & 2 \ 2 & 3 \ 3 & 4 \ \hline \end{array} \]

Calculating the ratios \( \frac{y}{x} \):

• For \( (1, 2) \): \( \frac{2}{1} = 2 \)
• For \( (2, 3) \): \( \frac{3}{2} = 1.5 \)
• For \( (3, 4) \): \( \frac{4}{3} \approx 1.33 \)

The ratios \( \frac{y}{x} \) are not the same for all pairs, so Function 2 is not proportional.

Thus, the only function that represents a proportional relationship is Function 1.

Answer 2

Determine if the given graph is the graph of the function y=−2/5x+3
. Enter 1 for yes. Enter 2 for no.
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.

Answer 3

To determine if the given graph represents the function \( y = -\frac{2}{5}x + 3 \), we can analyze the equation.

1. Identify the slope and y-intercept: The equation \( y = -\frac{2}{5}x + 3 \) has:

   • A slope \( m = -\frac{2}{5} \) (which indicates the line will go downwards as \( x \) increases).
   • A y-intercept \( b = 3 \) (the point where the line crosses the y-axis is \( (0, 3) \)).

2. Check the points given: The points mentioned in the graph are:

   • \( (0, 3) \): This matches the y-intercept.
   • \( (5, 1) \): We can check if this point lies on the line by substituting \( x = 5 \) into the function.

   \[ y = -\frac{2}{5}(5) + 3 = -2 + 3 = 1 \]

   Since substituting \( x = 5 \) gives \( y = 1 \), the point \( (5, 1) \) is also correct for this function.

3. Conclusion: Since the line accurately represents both the y-intercept and the slope, and the points \( (0, 3) \) and \( (5, 1) \) lie on the line described by the function, the graph is indeed the graph of the function \( y = -\frac{2}{5}x + 3 \).

Thus, you should enter 1 for yes.

Answer 4

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.

Answer 5

To determine the initial value and the rate of change in the description provided:

1. Initial Value: The initial value refers to the height of the tomato plant at the starting point of the observation, which in this case is today. Today, the height of the plant is given as 10 inches. Therefore, the initial value is: \[ \text{Initial Value} = 10 \text{ inches} \]

2. Rate of Change: The rate of change indicates how much the height of the plant increases over a given period of time. The plant grows at a rate of 1.7 inches per day. Therefore, the rate of change is: \[ \text{Rate of Change} = 1.7 \text{ inches per day} \]

To summarize:

• Initial Value: 10 inches
• Rate of Change: 1.7 inches per day

Answer 6

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.

Answer 7

To determine the rate of change and the initial value from the provided table of values, we need to analyze the changes in \(y\) as \(x\) changes.

Given Table:

x   y
2   22
8   25

Step 1: Determine the Rate of Change

The rate of change (slope, \(m\)) can be calculated using the formula:

\[ m = \frac{\Delta y}{\Delta x} = \frac{y_2 - y_1}{x_2 - x_1} \]

Using the points \( (x_1, y_1) = (2, 22) \) and \( (x_2, y_2) = (8, 25) \):

\[ m = \frac{25 - 22}{8 - 2} = \frac{3}{6} = \frac{1}{2} \]

Rate of Change: The rate of change is \( \frac{1}{2} \).

Step 2: Determine the Initial Value

The initial value refers to the value of \(y\) when \(x = 0\). Since the table does not directly provide \(y\) when \(x = 0\), we can use the linear function equation form, \(y = mx + b\).

Using any of the points and the slope, we can substitute \(m\) and one of the points to find \(b\) (the y-intercept):

Using the point \( (2, 22) \):

\[ 22 = \left(\frac{1}{2}\right)(2) + b \] \[ 22 = 1 + b \] \[ b = 22 - 1 = 21 \]

Thus, the equation of the linear function is:

\[ y = \frac{1}{2}x + 21 \]

From this equation, the initial value (the y-intercept when \(x = 0\)) is \(21\).

Summary:

• Rate of Change: \( \frac{1}{2} \) (or 0.5)
• Initial Value: \( 21 \)

Answer 8

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. alt="An illustration of a coordinate plane with four quadrants shows the x and y axes extending from negative 5 to 5 in increments of 1. A downward slanting line is plotted on the coordinate plane. The downward slanting line with arrows on both ends passes through the following closed points: left parenthesis negative 3 comma 4 right parenthesis, left parenthesis 0 comma 3 right parenthesis, left parenthesis 3 comma 2 right parenthesis, and left parenthesis 6 comma 1 right parenthesis. "

Answer 9

To determine the initial value and the rate of change of the linear function represented in the graph, we need to analyze the given points and use them to find the slope and the y-intercept of the line.

Step 1: Identify Points

From the description, we have the following points from the graph:

• \( (-3, 4) \)
• \( (0, 3) \)
• \( (3, 2) \)
• \( (6, 1) \)

Step 2: Find the Rate of Change (Slope)

The rate of change is given by the slope \( m \), which can be calculated using the formula:

\[ m = \frac{y_2 - y_1}{x_2 - x_1} \]

Let's use the points \( (0, 3) \) and \( (6, 1) \) to calculate the slope:

• Let \( (x_1, y_1) = (0, 3) \)
• Let \( (x_2, y_2) = (6, 1) \)

\[ m = \frac{1 - 3}{6 - 0} = \frac{-2}{6} = -\frac{1}{3} \approx -0.333 \]

Step 3: Determine the Initial Value (Y-intercept)

The initial value, or y-intercept, is the value of \( y \) when \( x = 0 \). From the points given, it is clear that when \( x = 0 \), \( y = 3 \).

Thus,

Initial Value (b): \( 3.000 \) (rounded to three decimal places).

Summary

• Rate of Change (Slope): \( -0.333 \) (rounded to three decimal places).
• Initial Value (Y-intercept): \( 3.000 \) (rounded to three decimal places).

To conclude:

• Rate of Change: \( -0.333 \)
• Initial Value: \( 3.000 \)