Question

Relationship B has a greater rate than Relationship A. This graph represents Relationship A.

Which equations could represent Relationship B?

Select each correct answer.

Responses

y=34x
y equals fraction numerator 3 over denominator 4 end fraction x

y=14x
y equals 1 fourth x

y=23x
y equals fraction numerator 2 over denominator 3 end fraction x

y = 0.6x
y = 0.6 x
First-quadrant graph showing a ray from the origin through the points begin ordered pair 4 comma 2 end ordered pair and begin ordered pair 8 comma 4 end ordered pair.

Answer 1

To determine which equations could represent Relationship B, we need to understand that Relationship B has a greater rate (slope) than Relationship A. From the description of Relationship A, we can deduce some information about its slope.

The points mentioned in Relationship A are (4, 2) and (8, 4). To find the slope (rate) of Relationship A, we can calculate the slope using the formula:

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

Using the points (4, 2) and (8, 4) as \((x_1, y_1)\) and \((x_2, y_2)\):

\[
\text{slope} = \frac{4 - 2}{8 - 4} = \frac{2}{4} = \frac{1}{2}
\]

So, the slope of Relationship A is \( \frac{1}{2} \).

Now, since Relationship B has a greater rate, we are looking for equations with a slope greater than \( \frac{1}{2} \).

Now let's evaluate the provided responses:

1. **\(y = 34x\)**: The slope is 34, which is greater than \( \frac{1}{2} \). **(Valid)**

2. **\(y = \frac{3}{4} x\)**: The slope is \( \frac{3}{4} \), which is greater than \( \frac{1}{2} \). **(Valid)**

3. **\(y = 14x\)**: The slope is 14, which is greater than \( \frac{1}{2} \). **(Valid)**

4. **\(y = \frac{1}{4} x\)**: The slope is \( \frac{1}{4} \), which is less than \( \frac{1}{2} \). **(Not valid)**

5. **\(y = \frac{2}{3} x\)**: The slope is \( \frac{2}{3} \), which is greater than \( \frac{1}{2} \). **(Valid)**

6. **\(y = 0.6 x\)**: The slope is 0.6, which is greater than \( \frac{1}{2} \). **(Valid)**

In summary, the equations that could represent Relationship B, having a greater rate than Relationship A, are:

- \(y = 34x\)
- \(y = \frac{3}{4} x\)
- \(y = 14x\)
- \(y = \frac{2}{3} x\)
- \(y = 0.6x\)