Which linear relationship can be described as a function with direct variation?A coordinate plane with a line passing through (negative 2, 3), (0, 1) and (1, 0).A coordinate plane with a line passing through (negative 2, negative 2), (0, 0) and (2, 2).A coordinate plane with a line passing through (negative 2, negative 1), (0, negative 1) and (2, negative 1).A coordinate plane with a line passing through (1, 1), (1, negative 1) and (1, negative 4).

Answer 1

A function exhibits direct variation if it can be represented in the form \( y = kx \), where \( k \) is a non-zero constant, meaning that the line passes through the origin (0, 0) and has a constant ratio of \( y \) to \( x \).

Let's analyze the given options:

1. The line passing through (negative 2, 3), (0, 1), and (1, 0):

   • This line does not pass through the origin (0, 0). It does not exhibit direct variation.

2. The line passing through (negative 2, negative 2), (0, 0), and (2, 2):

   • This line passes through the origin (0, 0) and is consistent with the properties of direct variation. It represents the form \( y = kx \) for some constant \( k \) (in this case, \( k = 1 \)).

3. The line passing through (negative 2, negative 1), (0, negative 1), and (2, negative 1):

   • This line is horizontal and does not pass through the origin. It is not a direct variation relationship.

4. The line passing through (1, 1), (1, negative 1), and (1, negative 4):

   • This line is vertical and does not represent a function (as it would fail the vertical line test). It does not exhibit direct variation.

Based on the analysis, the linear relationship that can be described as a function with direct variation is the line passing through (negative 2, negative 2), (0, 0), and (2, 2).

Answer 2

graph of the horizontal line that starts all the way to the top right corner

The conversion of square feet to square yards can be represented by direct variation. Three square yards are equivalent to 27 square feet. Let y represent the number of square yards and x the number of square feet. Then y varies directly with x.

What is the constant of variation?

What is the equation representing the direct variation?

Answer 3

To convert square feet to square yards, we know that 1 square yard is equivalent to 9 square feet (since 1 yard = 3 feet, thus \(1 \text{ square yard} = 3 \text{ feet} \times 3 \text{ feet} = 9 \text{ square feet}\)). Since 3 square yards are equivalent to 27 square feet, we can verify this relationship.

Finding the Constant of Variation

Since we have:

• \( y = \) square yards
• \( x = \) square feet

From the given relation:

• \( 3 \text{ square yards} = 27 \text{ square feet} \)

We can set this up using a direct variation formula: \[ y = kx \]

To find \( k \):

We will substitute the given values into the equation. Using \( y = 3 \) and \( x = 27 \): \[ 3 = k(27) \] \[ k = \frac{3}{27} = \frac{1}{9} \]

Conclusion:

• Constant of Variation \( k \): \( \frac{1}{9} \)

Finding the Equation of Direct Variation

Now that we have the constant of variation: \[ y = kx \] We can substitute \( k \): \[ y = \frac{1}{9}x \]

Conclusion:

• Equation representing the direct variation: \[ y = \frac{1}{9}x \]

In this context:

• \( y \) (in square yards) varies directly with \( x \) (in square feet) through the equation \( y = \frac{1}{9}x \).

Answer 4

What is the constant of variation?

✔ 1/9

What is the equation representing the direct variation?

✔ y = (1/9)x

Luis uses cubes to represent each term of a pattern based on a recursive function. The recursive function defined is f(n + 1) = f(n) + 4, where n is an integer and n ≥ 2. The number of cubes used in each of the first two figures is shown below. How many cubes does Luis use in the third, fourth, and fifth figures of the pattern? Fill in the blanks.

Figure 1: 9 cubes

Figure 2: 13 cubes

Figure 3:
cubes

Figure 4:
cubes

Figure 5:
cubes

Answer 5

The recursive function defined is \( f(n + 1) = f(n) + 4 \), which means that each term in the sequence can be found by adding 4 to the previous term.

To find the number of cubes used in each figure, we start with the first two figures:

• Figure 1: \( f(1) = 9 \) cubes
• Figure 2: \( f(2) = 13 \) cubes

Finding Figures 3, 4, and 5

Now, we apply the recursive function to find the next figures:

• Figure 3: \[ f(3) = f(2) + 4 = 13 + 4 = 17 \text{ cubes} \]

• Figure 4: \[ f(4) = f(3) + 4 = 17 + 4 = 21 \text{ cubes} \]

• Figure 5: \[ f(5) = f(4) + 4 = 21 + 4 = 25 \text{ cubes} \]

Summary of Cube Counts

• Figure 1: 9 cubes
• Figure 2: 13 cubes
• Figure 3: 17 cubes
• Figure 4: 21 cubes
• Figure 5: 25 cubes

Final Blanks Filled

• Figure 3: 17 cubes
• Figure 4: 21 cubes
• Figure 5: 25 cubes

Answer 6

The first five terms of an arithmetic sequence are shown below:

20, 17, 14, 11, 8, . . .

Let n represent the term number and f(n) the term in the sequence.

The function___
represents the sequence.

Answer 7

To represent the arithmetic sequence given by the first five terms:

20, 17, 14, 11, 8, ...

Step 1: Identify the first term and common difference

• The first term \( f(1) \) is 20.
• The common difference \( d \) can be found by subtracting the second term from the first term: \[ d = 17 - 20 = -3 \]

Step 2: Write the general formula for an arithmetic sequence

The formula for the \( n \)-th term of an arithmetic sequence can be expressed as: \[ f(n) = f(1) + (n - 1) \cdot d \] Substituting the known values:

• \( f(1) = 20 \)
• \( d = -3 \)

Step 3: Formulate the function

Substituting these values into the formula gives: \[ f(n) = 20 + (n - 1)(-3) \] Simplifying further: \[ f(n) = 20 - 3(n - 1) \] Expanding the equation: \[ f(n) = 20 - 3n + 3 \] Combining like terms: \[ f(n) = 23 - 3n \]

Final Representation

The function that represents the sequence is: \[ f(n) = 23 - 3n \]

Answer 8

not right
it must be f(n) = -3n + 23
f(n) = -3n + 61
f(n) = 20n - 3

Answer 9

Thank you for your patience. Let's correctly analyze the function that represents the arithmetic sequence given:

The first five terms of the arithmetic sequence are:

• 20, 17, 14, 11, 8

Step 1: Identify the first term and common difference.

• First term \( f(1) = 20 \)
• Common difference \( d = 17 - 20 = -3 \)

Step 2: For an arithmetic sequence, the general formula for the \( n \)-th term is: \[ f(n) = f(1) + (n - 1)d \] Where \( f(1) \) is the first term and \( d \) is the common difference.

Substituting the known values: \[ f(n) = 20 + (n - 1)(-3) \] This can be expanded as: \[ f(n) = 20 - 3(n - 1) \] \[ f(n) = 20 - 3n + 3 \] \[ f(n) = 23 - 3n \]

Thus, the correct function forms to match your context should be adjusted.

Given Options for the Function:

1. \( f(n) = -3n + 23 \)
2. \( f(n) = -3n + 61 \)
3. \( f(n) = 20n - 3 \)

Comparing our derived function:

1. Option 1: \( f(n) = -3n + 23 \) matches with \( f(n) = 23 - 3n \).
2. Option 2: \( f(n) = -3n + 61 \) does not correspond as it provides an incorrect value.
3. Option 3: \( f(n) = 20n - 3 \) does not fit.

Conclusion:

The correct representation of the arithmetic sequence should be: \[ f(n) = -3n + 23 \]

Please let me know if you have any other questions, or need further clarification!

Answer 10

A coordinate plane with Number of Text Messages on the x-axis and Number of Minutes Used on the y-axis, with points plotted at, (10, 20), (20, 40), (30, 60), and (40, 80).A prepaid cell phone charges a preset number of minutes to use text messaging. The graph represents y, the number of minutes used for x, the number of text messages sent and received. Is there a direct variation? Which equation represents the relationship?Yes, y = 2x.Yes, y = 20x.No, y = x + 10.No, y = x + 20.

Answer 11

To determine if there is a direct variation between the number of text messages (\(x\)) and the number of minutes used (\(y\)), we need to examine the points plotted on the coordinate plane and see if they fit the form \(y = kx\), where \(k\) is a constant.

Given Points:

1. (10, 20)
2. (20, 40)
3. (30, 60)
4. (40, 80)

Step 1: Identify the relationship

We can check if the ratio of \(y\) to \(x\) remains constant:

1. For (10, 20): \[ k = \frac{y}{x} = \frac{20}{10} = 2 \]
2. For (20, 40): \[ k = \frac{40}{20} = 2 \]
3. For (30, 60): \[ k = \frac{60}{30} = 2 \]
4. For (40, 80): \[ k = \frac{80}{40} = 2 \]

Since the ratio \(k\) is consistent at 2 for all the points, this indicates a direct variation.

Step 2: Write the Equation

Given that \(k = 2\), we can represent the direct variation as: \[ y = 2x \]

Conclusion

Yes, there is a direct variation, and the equation that represents the relationship is: \[ y = 2x \]

Thus, the correct answer is: Yes, \(y = 2x\).