Question

4* the line $y = \frac{1}{3}x + 7$ is graphed
match each graph characteristic with its value.
\tdomain
\trange
\tx - intercept
\ty - intercept
\tinterval of increasing
\tthe graph is positive.
\tthe graph is negative.

Explanation:

Step1: Analyze the line equation \( y = \frac{1}{3}x + 7 \)

This is a linear equation in slope - intercept form \( y=mx + b \), where \( m=\frac{1}{3} \) (slope) and \( b = 7 \) (y - intercept). The domain of a linear function (without any restrictions) is all real numbers. But looking at the graph (the line segment), we can see the x - values it covers. From the graph, the left - most point has \( x=-9 \) and the right - most point has \( x = 3 \), so the domain is \( [-9,3] \).

Step2: Find the range

For a linear function with a positive slope ( \( m=\frac{1}{3}>0 \) ), as \( x \) increases, \( y \) increases. When \( x=-9 \), \( y=\frac{1}{3}(-9)+7=- 3 + 7 = 4 \). When \( x = 3 \), \( y=\frac{1}{3}(3)+7=1 + 7 = 8 \). So the range is \( [4,8] \).

Step3: Find the x - intercept

The x - intercept is the value of \( x \) when \( y = 0 \). Set \( y = 0 \) in \( y=\frac{1}{3}x + 7 \):
\[ \begin{align*} 0&=\frac{1}{3}x+7\\ \frac{1}{3}x&=-7\\ x&=-21 \end{align*} \]
But from the graph, the line segment does not reach \( x=-21 \). Wait, maybe we made a mistake. Wait, looking at the graph, the line crosses the x - axis? Wait, no, the line \( y=\frac{1}{3}x + 7 \) (the segment) - wait, maybe the graph is a segment. Wait, when we look at the graph, the left end is at \( x=-9,y = 4 \) and right end at \( x = 3,y = 8 \). Wait, maybe the original problem's graph is a segment. Wait, maybe we should re - evaluate. Wait, the y - intercept: when \( x = 0 \), \( y=7 \). Let's check the graph: when \( x = 0 \), \( y = 7 \), so the y - intercept is 7.

Step4: Find the y - intercept

The y - intercept is the value of \( y \) when \( x = 0 \). Substitute \( x = 0 \) into \( y=\frac{1}{3}x+7 \), we get \( y = 7 \).

Step5: Interval of increasing

Since the slope \( m=\frac{1}{3}>0 \), the function is increasing on its domain. The domain of the segment is \( [-9,3] \), so the interval of increasing is \( [-9,3] \).

Step6: The graph is positive

A graph is positive when \( y>0 \). For the line \( y=\frac{1}{3}x + 7 \), when is \( y>0 \)? \( \frac{1}{3}x+7>0\Rightarrow x>- 21 \). But for the segment, since the minimum \( y \) - value on the segment is 4 (when \( x=-9 \)) and maximum is 8 (when \( x = 3 \)), and \( 4>0 \), the graph is positive on \( [-9,3] \).

Step7: The graph is negative

A graph is negative when \( y<0 \). But on the segment, the minimum \( y \) - value is 4, which is greater than 0, so the graph is never negative on the segment (the interval where the graph is negative would be \( (-\infty,-21) \), but the segment is not in that interval).

Domain:

The x - values of the line segment in the graph range from \( x=-9 \) to \( x = 3 \), so domain is \( [-9,3] \)

Range:

The y - values of the line segment range from \( y = 4 \) (when \( x=-9 \)) to \( y = 8 \) (when \( x = 3 \)), so range is \( [4,8] \)

x - intercept:

If we consider the entire line \( y=\frac{1}{3}x + 7 \), \( x=-21 \), but the segment does not reach there. Wait, maybe the graph is a segment, so maybe the x - intercept of the segment? But the segment does not cross the x - axis. Wait, maybe there is a mistake. Wait, the line \( y=\frac{1}{3}x + 7 \) has a y - intercept at \( (0,7) \), which matches the graph.

y - intercept:

When \( x = 0 \), \( y = 7 \), so the y - intercept is 7.

Interval of increasing:

Since the slope is positive, the function is increasing on its domain \( [-9,3] \), so the interval of increasing is \( [-9,3] \)

The graph is positive:

The y - values of the segment are from 4 to 8, all greater than 0, so the interval where the graph is positive is \( [-9,3] \)

The graph is negative:

The y - values of the segment are all greater than 0, so there is no interval where the graph is negative (or the interval is empty)

Now, let's match:

- Domain: \( [-9,3] \)
- Range: \( [4,8] \)
- x - intercept: - 21 (if we consider the entire line) or no x - intercept on the segment. But maybe the problem considers the entire line? Wait, the original equation is \( y=\frac{1}{3}x + 7 \), so for the entire line, x - intercept is - 21, y - intercept is 7, domain \( (-\infty,\infty) \), range \( (-\infty,\infty) \), but the graph is a segment. This is a bit confusing. Wait, maybe the graph is the line segment from \( (-9,4) \) to \( (3,8) \). Let's re - calculate:

For the segment:

- Domain: \( x\in[-9,3] \)
- Range: \( y\in[4,8] \)
- x - intercept: The segment does not cross the x - axis (since when \( x=-9,y = 4>0 \) and as \( x \) increases, \( y \) increases), so no x - intercept on the segment. But the line \( y=\frac{1}{3}x + 7 \) has x - intercept at \( x=-21 \)
- y - intercept: At \( x = 0 \), \( y = 7 \) (which is on the segment, since \( 0\in[-9,3] \) and \( 7\in[4,8] \))
- Interval of increasing: Since slope is positive, on \( [-9,3] \)
- The graph is positive: On \( [-9,3] \) (since \( y\geq4>0 \))
- The graph is negative: Nowhere on \( [-9,3] \) (since \( y\geq4>0 \))

Let's assume the problem is about the line (not the segment) for the intercepts:

- Domain (of the line): All real numbers \( (-\infty,\infty) \)
- Range (of the line): All real numbers \( (-\infty,\infty) \)
- x - intercept: \( x=-21 \) (when \( y = 0 \))
- y - intercept: \( y = 7 \) (when \( x = 0 \))
- Interval of increasing: \( (-\infty,\infty) \) (since slope is positive)
- The graph is positive: \( x>-21 \) (since \( y=\frac{1}{3}x + 7>0\Rightarrow x>-21 \))
- The graph is negative: \( x<-21 \) (since \( y=\frac{1}{3}x + 7<0\Rightarrow x<-21 \))

But since there is a graph (a segment), we should go with the segment:

- Domain: \( [-9,3] \)
- Range: \( [4,8] \)
- x - intercept: None (on the segment)
- y - intercept: 7
- Interval of increasing: \( [-9,3] \)
- The graph is positive: \( [-9,3] \)
- The graph is negative: No interval (or \( \varnothing \))

Let's list the matches:

- Domain: \( [-9,3] \)
- Range: \( [4,8] \)
- x - intercept: - 21 (if line) or none (if segment)
- y - intercept: 7
- Interval of increasing: \( [-9,3] \) (if segment) or \( (-\infty,\infty) \) (if line)
- The graph is positive: \( [-9,3] \) (if segment) or \( (-21,\infty) \) (if line)
- The graph is negative: \( \varnothing \) (if segment) or \( (-\infty,-21) \) (if line)

Assuming the problem expects the values for the line (not the segment) for intercepts:

- Domain: All real numbers ( \( (-\infty, \infty) \) )
- Range: All real numbers ( \( (-\infty, \infty) \) )
- x - intercept: - 21
- y - intercept: 7
- Interval of increasing: \( (-\infty, \infty) \)
- The graph is positive: \( (-21, \infty) \)
- The graph is negative: \( (-\infty, -21) \)

If we consider the segment:

- Domain: \( [-9, 3] \)
- Range: \( [4, 8] \)
- x - intercept: None (on the segment)
- y - intercept: 7
- Interval of increasing: \( [-9, 3] \)
- The graph is positive: \( [-9, 3] \)
- The graph is negative: No interval

Since the graph is shown with a segment from \( x=-9 \) to \( x = 3 \), we will use the segment values:

1. Domain: \( [-9,3] \)
2. Range: \( [4,8] \)
3. x - intercept: (No x - intercept on the segment, but if we consider the line, - 21)
4. y - intercept: 7
5. Interval of increasing: \( [-9,3] \)
6. The graph is positive: \( [-9,3] \)
7. The graph is negative: (No interval on the segment)

Let's match them properly:

- Domain: \( [-9, 3] \)
- Range: \( [4, 8] \)
- x - intercept: - 21 (if we use the line equation)
- y - intercept: 7
- Interval of increasing: \( [-9, 3] \) (since the segment is increasing)
- The graph is positive: \( [-9, 3] \) (since all y - values in the segment are positive)
- The graph is negative: (No interval in the segment)

So the matches are:

- domain: \( [-9, 3] \)
- range: \( [4, 8] \)
- x - intercept: - 21
- y - intercept: 7
- interval of increasing: \( [-9, 3] \)
- The graph is positive: \( [-9, 3] \)
- The graph is negative: (No interval, or \( (-\infty, - 21) \) if considering the line)

Answer:

• domain: \(\boldsymbol{[-9, 3]}\)
• range: \(\boldsymbol{[4, 8]}\)
• x - intercept: \(\boldsymbol{-21}\)
• y - intercept: \(\boldsymbol{7}\)
• interval of increasing: \(\boldsymbol{[-9, 3]}\)
• The graph is positive: \(\boldsymbol{[-9, 3]}\)
• The graph is negative: \(\boldsymbol{(-\infty, - 21)}\) (if considering the line) or \(\boldsymbol{\varnothing}\) (if considering the segment)