Question

Function
\[1\] is defined by the equation

\[y=-\dfrac{7}{3}x-1\].
Function
\[2\] is defined by the following table.
\[x\]
\[y\]
\[0\]
\[-2\]
\[3\]
\[-8\]
\[7\]
\[-16\]
\[11\]
\[-24\]
Which function has a more negative slope?
Choose 1 answer:
Choose 1 answer:
(Choice A) Function
\[1\]
A
Function
\[1\]
(Choice B) Function
\[2\]
B
Function
\[2\]
(Choice C) The functions have the same slope.
C
The functions have the same slope.

Answer 1

To determine which function has a more negative slope, we first need to find the slope of each function.

**Function 1:**
The equation is given as:
\[
y = -\frac{7}{3}x - 1
\]
From this equation, we can see that the slope (the coefficient of \(x\)) is:
\[
m_1 = -\frac{7}{3}
\]

**Function 2:**
We need to calculate the slope based on the values provided in the table. The slope is determined by finding the change in \(y\) for a change in \(x\) between the pairs of points provided.

Let's calculate the slope between the points:

1. Between \((0, -2)\) and \((3, -8)\):
\[
m = \frac{-8 - (-2)}{3 - 0} = \frac{-8 + 2}{3} = \frac{-6}{3} = -2
\]

2. Between \((3, -8)\) and \((7, -16)\):
\[
m = \frac{-16 - (-8)}{7 - 3} = \frac{-16 + 8}{4} = \frac{-8}{4} = -2
\]

3. Between \((7, -16)\) and \((11, -24)\):
\[
m = \frac{-24 - (-16)}{11 - 7} = \frac{-24 + 16}{4} = \frac{-8}{4} = -2
\]

From these calculations, we see that the slope of Function 2 is:
\[
m_2 = -2
\]

Now, let's compare the slopes of the two functions:

- Function 1: \(m_1 = -\frac{7}{3} \approx -2.33\)
- Function 2: \(m_2 = -2\)

Now comparing \(-\frac{7}{3}\) and \(-2\):
Since \(-\frac{7}{3} \approx -2.33\) is more negative than \(-2\), we conclude that:

**Answer:** (Choice A) Function \(1\) has a more negative slope.