Question
Which linear function has the steepest slope?
On a coordinate plane, a line goes through points (0, 3) and (4, 2).
y = negative 0.1 x minus 5
A 2-column table with 5 rows. Column 1 is labeled x with entries negative 4, negative 2, 0, 2, 4. Column 2 is labeled y with entries 0, 0.5, 1.0, 1.5, 2.0.
A 2-column table with 5 rows. Column 1 is labeled x with entries negative 6, negative 2, 1, 7, 9. Column 2 is labeled y with entries 6.6, 4.2, 2.4, negative 1.2, negative 2.4.
On a coordinate plane, a line goes through points (0, 3) and (4, 2).
y = negative 0.1 x minus 5
A 2-column table with 5 rows. Column 1 is labeled x with entries negative 4, negative 2, 0, 2, 4. Column 2 is labeled y with entries 0, 0.5, 1.0, 1.5, 2.0.
A 2-column table with 5 rows. Column 1 is labeled x with entries negative 6, negative 2, 1, 7, 9. Column 2 is labeled y with entries 6.6, 4.2, 2.4, negative 1.2, negative 2.4.
Answers
GPT-4o mini
To determine which linear function has the steepest slope, we first need to calculate the slope of each line provided.
1. **Line through points (0, 3) and (4, 2):**
The slope \( m \) is given by the formula:
\[
m = \frac{y_2 - y_1}{x_2 - x_1}
\]
Plugging in the points (0, 3) and (4, 2):
\[
m = \frac{2 - 3}{4 - 0} = \frac{-1}{4} = -0.25
\]
2. **Line represented by the function \( y = -0.1x - 5 \):**
The slope of this line is simply the coefficient in front of \( x \), which is \( -0.1 \).
3. **From the first table \( (x, y) \):**
Assuming the points derived from the table (x: -4, -2, 0, 2, 4 with corresponding y values) make a linear function:
- Pair of points: (-4, 0) and (4, 2)
\[
m = \frac{2 - 0}{4 - (-4)} = \frac{2}{8} = \frac{1}{4} = 0.25
\]
4. **From the second table \( (x, y) \):**
We will check points (on the extremes):
- Pair of points: (-6, 6.6) and (9, -2.4)
\[
m = \frac{-2.4 - 6.6}{9 - (-6)} = \frac{-9}{15} = -0.6
\]
Now we compare the slopes:
- Slope from points (0, 3) and (4, 2): \( -0.25 \)
- Slope from \( y = -0.1x - 5 \): \( -0.1 \)
- Slope from the first table: \( 0.25 \)
- Slope from the second table: \( -0.6 \)
The steepest slope is the one with the greatest absolute value. Thus, the function represented by the first table \( (x, y) \) with a slope of \( 0.25 \), which is positive, is the steepest among the given functions.
So, the linear function with the steepest slope is the one derived from **the first table**, whose slope is \( 0.25 \).
1. **Line through points (0, 3) and (4, 2):**
The slope \( m \) is given by the formula:
\[
m = \frac{y_2 - y_1}{x_2 - x_1}
\]
Plugging in the points (0, 3) and (4, 2):
\[
m = \frac{2 - 3}{4 - 0} = \frac{-1}{4} = -0.25
\]
2. **Line represented by the function \( y = -0.1x - 5 \):**
The slope of this line is simply the coefficient in front of \( x \), which is \( -0.1 \).
3. **From the first table \( (x, y) \):**
Assuming the points derived from the table (x: -4, -2, 0, 2, 4 with corresponding y values) make a linear function:
- Pair of points: (-4, 0) and (4, 2)
\[
m = \frac{2 - 0}{4 - (-4)} = \frac{2}{8} = \frac{1}{4} = 0.25
\]
4. **From the second table \( (x, y) \):**
We will check points (on the extremes):
- Pair of points: (-6, 6.6) and (9, -2.4)
\[
m = \frac{-2.4 - 6.6}{9 - (-6)} = \frac{-9}{15} = -0.6
\]
Now we compare the slopes:
- Slope from points (0, 3) and (4, 2): \( -0.25 \)
- Slope from \( y = -0.1x - 5 \): \( -0.1 \)
- Slope from the first table: \( 0.25 \)
- Slope from the second table: \( -0.6 \)
The steepest slope is the one with the greatest absolute value. Thus, the function represented by the first table \( (x, y) \) with a slope of \( 0.25 \), which is positive, is the steepest among the given functions.
So, the linear function with the steepest slope is the one derived from **the first table**, whose slope is \( 0.25 \).