Question

the line ( y = -2x - 4 ) is graphed
match each graph characteristic with its value

• domain
• range
• x - intercept(s)
• y - intercept(s)
• the graph is positive
• the graph is negative

options: ( (-4,0) ), ( (-2,8) ), ( (0,-4) ), ( (-2,0) ), ( {x|x < -2} ), ( {x|x > -2} ), ( {x|xin mathbb{z}} ), ( {x|xin mathbb{r}} ), ( {y|yin mathbb{z}} ), ( {y|yin mathbb{r}} )

Explanation:

Step1: Analyze Domain

The graph is a line segment (from the grid and the line \(y = -2x - 4\) context). To find the domain, we look at the x - values of the endpoints. From the graph (assuming the left endpoint is \(x=-2\) and right endpoint \(x = 2\)? Wait, no, let's re - examine. Wait, the line \(y=-2x - 4\). Let's find the x - intercept: set \(y = 0\), \(0=-2x-4\), \(2x=-4\), \(x=-2\). Y - intercept: set \(x = 0\), \(y=-4\). But the graph is a segment. Wait, the domain of a graph (the set of x - values it covers). Looking at the grid, the horizontal extent: from \(x=-2\) to \(x = 2\)? Wait, no, the options have \(\{x|x\in R\}\) (no, that's for a line, but this is a segment). Wait, the options for domain: wait, the first column is "domain", "range", "x - intercept", "y - intercept", "The graph is positive", "The graph is negative". The right - hand options: \([-4,0]\), \((-2,2)\)? No, wait, the x - intercept is when \(y = 0\), \(y=-2x - 4=0\) gives \(x=-2\), but the options for x - intercept: \((-2,0)\)? Wait, let's match each:

1. **Domain**: The domain is the set of x - values. If the graph is a line segment (from the grid, the horizontal span). Wait, the line \(y=-2x - 4\) as a segment: let's assume the endpoints are at \(x=-2\) (left) and \(x = 2\) (right)? No, the options for domain: one of the right - hand options is \(\{x|x\in R\}\) (no, that's for a line, but maybe it's a segment? Wait, no, maybe the graph is a line (the equation is \(y=-2x - 4\), a linear function). Wait, linear functions have domain \(\{x|x\in R\}\) (all real numbers) because there are no restrictions on x for a linear function \(y = mx + b\). So domain matches \(\{x|x\in R\}\).

2. **Range**: The range of \(y=-2x - 4\) is also \(\{y|y\in R\}\) if it's a line, but the option \([-4,0]\) is a range. Wait, maybe the graph is a segment. Let's find the range: when \(x=-2\), \(y=-2(-2)-4 = 4 - 4=0\); when \(x = 0\), \(y=-4\); when \(x = 2\), \(y=-2(2)-4=-4 - 4=-8\)? No, that doesn't match. Wait, maybe the graph is a triangle? Wait, the grid has a red line from top - left to bottom - right. Let's re - evaluate:

- **x - intercept**: The x - intercept is the point where \(y = 0\). For \(y=-2x - 4\), \(0=-2x-4\Rightarrow x=-2\), so the x - intercept is \((-2,0)\).

- **y - intercept**: The y - intercept is where \(x = 0\), \(y=-2(0)-4=-4\), so the y - intercept is \((0,-4)\).

- **Domain**: For a linear function \(y=-2x - 4\), the domain is all real numbers, so \(\{x|x\in R\}\).

- **Range**: For a linear function with \(m=-2 eq0\), the range is all real numbers, so \(\{y|y\in R\}\). But one of the range options is \([-4,0]\). Wait, maybe the graph is a segment (not the entire line). Let's assume the graph is a segment from \(x=-2\) (where \(y = 0\)) to \(x = 0\) (where \(y=-4\))? No, the grid shows a longer segment. Wait, let's match the given options:

- **Domain**: The set of x - values. If the graph is a line (the equation is linear), domain is \(\{x|x\in R\}\).

- **Range**: The set of y - values. If it's a line, range is \(\{y|y\in R\}\), but one of the range options is \([-4,0]\). Wait, maybe the graph is a segment between two points. Let's find two points on \(y=-2x - 4\): when \(x=-2\), \(y = 0\); when \(x = 0\), \(y=-4\); when \(x = 2\), \(y=-8\). No, the grid has a red line from top (high y) to bottom (low y). Wait, the range option \([-4,0]\): if the graph goes from \(y=-4\) to \(y = 0\), that's a range of \([-4,0]\).

- **x - intercept**: The x - intercept is \((-2,0)\) (since \(y = 0\) at \(x=-2\)).

- **y - intercept**: The y - intercept is \((0,-4)\) (since \(x = 0\) at \(y=-4\)).

- **The graph is positive**: The graph is positive when \(y>0\). Solve \(y=-2x - 4>0\), \(-2x>4\), \(x<-2\). So the solution is \(\{x|x < - 2\}\).

- **The graph is negative**: The graph is negative when \(y<0\). Solve \(y=-2x - 4<0\), \(-2x<4\), \(x>-2\). So the solution is \(\{x|x > - 2\}\).

Now let's match each:

- **Domain**: A linear function (the equation \(y=-2x - 4\)) has domain all real numbers, so domain \(\longleftrightarrow\{x|x\in R\}\)

- **Range**: If the graph is a segment (from the grid, maybe between \(y=-4\) and \(y = 0\)), so range \(\longleftrightarrow[-4,0]\)

- **x - intercept**: The x - intercept is \((-2,0)\) (since \(y = 0\) at \(x=-2\)), so x - intercept \(\longleftrightarrow(-2,0)\)

- **y - intercept**: The y - intercept is \((0,-4)\) (since \(x = 0\) at \(y=-4\)), so y - intercept \(\longleftrightarrow(0,-4)\)

- **The graph is positive**: Solve \(y=-2x - 4>0\), \(x<-2\), so \(\{x|x < - 2\}\)

- **The graph is negative**: Solve \(y=-2x - 4<0\), \(x>-2\), so \(\{x|x > - 2\}\)

Step - by - Step Matching:

1. **Domain**:
- A linear function \(y = mx + b\) (here \(m=-2\), \(b = - 4\)) has no restrictions on the input \(x\). So the domain is all real numbers, \(\{x|x\in R\}\). So domain \(\longleftrightarrow\{x|x\in R\}\)

2. **Range**:
- For the line \(y=-2x - 4\), if we consider the graph (from the grid, the vertical span). If the graph goes from \(y=-4\) (lowest) to \(y = 0\) (highest), the range is \([-4,0]\). So range \(\longleftrightarrow[-4,0]\)

3. **x - intercept**:
- The x - intercept is the point where \(y = 0\). Set \(y = 0\) in \(y=-2x - 4\): \(0=-2x-4\Rightarrow2x=-4\Rightarrow x=-2\). So the x - intercept is \((-2,0)\). So x - intercept \(\longleftrightarrow(-2,0)\)

4. **y - intercept**:
- The y - intercept is the point where \(x = 0\). Set \(x = 0\) in \(y=-2x - 4\): \(y=-2(0)-4=-4\). So the y - intercept is \((0,-4)\). So y - intercept \(\longleftrightarrow(0,-4)\)

5. **The graph is positive**:
- The graph is positive when \(y>0\). So solve \(y=-2x - 4>0\).
- Add \(2x\) to both sides: \(0 + 2x>-2x-4 + 2x\Rightarrow2x>-4\) (no, wait, original inequality: \(y=-2x - 4>0\)
- Add 4 to both sides: \(-2x>4\)
- Divide both sides by \(-2\) (and reverse the inequality sign): \(x<-2\). So the solution is \(\{x|x < - 2\}\). So "The graph is positive" \(\longleftrightarrow\{x|x < - 2\}\)

6. **The graph is negative**:
- The graph is negative when \(y<0\). Solve \(y=-2x - 4<0\).
- Add \(2x\) to both sides: \(y + 2x<-4+2x\) (no, better: \(y=-2x - 4<0\)
- Add 4 to both sides: \(-2x<4\)
- Divide both sides by \(-2\) (reverse inequality): \(x>-2\). So the solution is \(\{x|x > - 2\}\). So "The graph is negative" \(\longleftrightarrow\{x|x > - 2\}\)

Answer:

• Domain \(\longleftrightarrow\{x|x\in R\}\)
• Range \(\longleftrightarrow[-4,0]\)
• x - intercept \(\longleftrightarrow(-2,0)\)
• y - intercept \(\longleftrightarrow(0,-4)\)
• The graph is positive \(\longleftrightarrow\{x|x < - 2\}\)
• The graph is negative \(\longleftrightarrow\{x|x > - 2\}\)