Question
Compare the linear functions expressed below by data in a table and by an equation.
A 2-column table with 4 rows. Column 1 is labeled x with entries negative 6, negative four-thirds, 1, 4. Column 2 is labeled y with entries negative StartFraction 9 Over 2 EndFraction, negative 1, three-fourths, 3. y = three-fourths x.
Which of the following statements are true? Select all that apply.
If the equation were graphed, it would be a horizontal line.
Both functions have the same slope.
The origin is the y-intercept for the function expressed in the table.
The linear equation does not have a y-intercept.
The table and the graph express an equivalent function.
A 2-column table with 4 rows. Column 1 is labeled x with entries negative 6, negative four-thirds, 1, 4. Column 2 is labeled y with entries negative StartFraction 9 Over 2 EndFraction, negative 1, three-fourths, 3. y = three-fourths x.
Which of the following statements are true? Select all that apply.
If the equation were graphed, it would be a horizontal line.
Both functions have the same slope.
The origin is the y-intercept for the function expressed in the table.
The linear equation does not have a y-intercept.
The table and the graph express an equivalent function.
Answers
GPT-4o mini
Let's analyze the linear functions based on the provided information.
1. **Equation given**: \( y = \frac{3}{4}x \)
- From the equation, the slope (m) is \( \frac{3}{4} \) and the y-intercept (b) is \( 0 \). Thus, the line passes through the origin (0,0).
2. **Data from the table**:
- \((-6, -\frac{9}{2})\)
- \((- \frac{4}{3}, -1)\)
- \((1, \frac{3}{4})\)
- \((4, 3)\)
To see if these points fit the same linear function, we can check the slope between each pair of points:
- For the points (-6, -4.5) and (1, 0.75):
\[
\text{slope} = \frac{0.75 - (-4.5)}{1 - (-6)} = \frac{0.75 + 4.5}{1 + 6} = \frac{5.25}{7} = \frac{3}{4}
\]
- For points (1, 0.75) and (4, 3):
\[
\text{slope} = \frac{3 - 0.75}{4 - 1} = \frac{2.25}{3} = \frac{3}{4}
\]
Through both calculations, we see that the slope between the points is consistently \( \frac{3}{4} \).
Now, let’s evaluate the provided statements:
1. **If the equation were graphed, it would be a horizontal line.**
- This statement is **false** because the slope is \( \frac{3}{4} \), which indicates an incline, not a horizontal line.
2. **Both functions have the same slope.**
- This statement is **true** since we determined the slope from the table matches that of the equation, which is \( \frac{3}{4} \).
3. **The origin is the y-intercept for the function expressed in the table.**
- This statement is **false** because there is no point in the table where \( x = 0 \). Moreover, since there’s no entry for \( (0, 0) \) in the table, the y-intercept cannot be confidently established.
4. **The linear equation does not have a y-intercept.**
- This statement is **false** because every linear function expressed in the form \( y = mx + b \) has a y-intercept (in this case, \( b = 0 \)).
5. **The table and the graph express an equivalent function.**
- This statement is **true** since the table points all pass through the same linear relationship defined by \( y = \frac{3}{4}x \).
**In summary, the true statements are:**
- Both functions have the same slope.
- The table and the graph express an equivalent function.
1. **Equation given**: \( y = \frac{3}{4}x \)
- From the equation, the slope (m) is \( \frac{3}{4} \) and the y-intercept (b) is \( 0 \). Thus, the line passes through the origin (0,0).
2. **Data from the table**:
- \((-6, -\frac{9}{2})\)
- \((- \frac{4}{3}, -1)\)
- \((1, \frac{3}{4})\)
- \((4, 3)\)
To see if these points fit the same linear function, we can check the slope between each pair of points:
- For the points (-6, -4.5) and (1, 0.75):
\[
\text{slope} = \frac{0.75 - (-4.5)}{1 - (-6)} = \frac{0.75 + 4.5}{1 + 6} = \frac{5.25}{7} = \frac{3}{4}
\]
- For points (1, 0.75) and (4, 3):
\[
\text{slope} = \frac{3 - 0.75}{4 - 1} = \frac{2.25}{3} = \frac{3}{4}
\]
Through both calculations, we see that the slope between the points is consistently \( \frac{3}{4} \).
Now, let’s evaluate the provided statements:
1. **If the equation were graphed, it would be a horizontal line.**
- This statement is **false** because the slope is \( \frac{3}{4} \), which indicates an incline, not a horizontal line.
2. **Both functions have the same slope.**
- This statement is **true** since we determined the slope from the table matches that of the equation, which is \( \frac{3}{4} \).
3. **The origin is the y-intercept for the function expressed in the table.**
- This statement is **false** because there is no point in the table where \( x = 0 \). Moreover, since there’s no entry for \( (0, 0) \) in the table, the y-intercept cannot be confidently established.
4. **The linear equation does not have a y-intercept.**
- This statement is **false** because every linear function expressed in the form \( y = mx + b \) has a y-intercept (in this case, \( b = 0 \)).
5. **The table and the graph express an equivalent function.**
- This statement is **true** since the table points all pass through the same linear relationship defined by \( y = \frac{3}{4}x \).
**In summary, the true statements are:**
- Both functions have the same slope.
- The table and the graph express an equivalent function.