Question
Select all graphs that the statement is true for.
A statement may be true for more than one graph.
Title linear graph with y-intercept at (0,2) and another point at (1, 5) quadratic graph with vertex at (4, -4) and x-intercepts at (2, 0) and (6, 0) exponential graph with y-intercept a 3, and asymptote at y = 2 absolute value graph with vertex at (3, 1) and x-intercepts at (2, 0) and (4, 0)
Domain: −∞≤x≤∞
Range: −∞≤y≤∞
Range: y≤1
Range: y>2
Range: y≥−4
y-intercept: (0, 2)
y-intercept: (0, 3)
A statement may be true for more than one graph.
Title linear graph with y-intercept at (0,2) and another point at (1, 5) quadratic graph with vertex at (4, -4) and x-intercepts at (2, 0) and (6, 0) exponential graph with y-intercept a 3, and asymptote at y = 2 absolute value graph with vertex at (3, 1) and x-intercepts at (2, 0) and (4, 0)
Domain: −∞≤x≤∞
Range: −∞≤y≤∞
Range: y≤1
Range: y>2
Range: y≥−4
y-intercept: (0, 2)
y-intercept: (0, 3)
Answers
GPT-4o mini
Based on the descriptions provided for each graph and the specified properties about their domains, ranges, and y-intercepts, we can analyze each graph:
1. **Linear graph with y-intercept at (0, 2) and another point at (1, 5)**:
- Domain: \(−∞ ≤ x ≤ ∞\) - **True**
- Range: \(−∞ ≤ y ≤ ∞\) - **True**
- Range: \(y ≤ 1\) - **False**
- Range: \(y > 2\) - **False**
- Range: \(y ≥ −4\) - **True**
- y-intercept: (0, 2) - **True**
- y-intercept: (0, 3) - **False**
2. **Quadratic graph with vertex at (4, -4) and x-intercepts at (2, 0) and (6, 0)**:
- Domain: \(−∞ ≤ x ≤ ∞\) - **True**
- Range: \(−∞ ≤ y ≤ ∞\) - **True**
- Range: \(y ≤ 1\) - **False**
- Range: \(y > 2\) - **False**
- Range: \(y ≥ −4\) - **True**
- y-intercept: (0, 2) - **False**
- y-intercept: (0, 3) - **False**
3. **Exponential graph with y-intercept at 3, and asymptote at y = 2**:
- Domain: \(−∞ ≤ x ≤ ∞\) - **True**
- Range: \(−∞ ≤ y ≤ ∞\) - **False** (Exponential graphs typically do not have an infinite range in the negative direction)
- Range: \(y ≤ 1\) - **False**
- Range: \(y > 2\) - **True** (since the asymptote at y = 2 means the graph approaches this value but does not cross it)
- Range: \(y ≥ −4\) - **True**
- y-intercept: (0, 2) - **False**
- y-intercept: (0, 3) - **True**
4. **Absolute value graph with vertex at (3, 1) and x-intercepts at (2, 0) and (4, 0)**:
- Domain: \(−∞ ≤ x ≤ ∞\) - **True**
- Range: \(−∞ ≤ y ≤ ∞\) - **False** (absolute value graphs typically have a minimum value)
- Range: \(y ≤ 1\) - **False**
- Range: \(y > 2\) - **False**
- Range: \(y ≥ −4\) - **True**
- y-intercept: (0, 2) - **False**
- y-intercept: (0, 3) - **False**
### Summary of findings:
- **Linear graph**: True for the domain, both ranges, and the y-intercept at (0, 2).
- **Quadratic graph**: True for the domain, both ranges (including y ≥ −4).
- **Exponential graph**: True for the domain, the range \(y > 2\), the range \(y ≥ −4\), and the y-intercept at (0, 3).
- **Absolute value graph**: True for the domain and the range \(y ≥ −4\).
### The graphs for which each statement is true:
- Domain: **All graphs**
- Range: \(−∞ ≤ y ≤ ∞\): **Linear graph, Quadratic graph**
- Range: \(y ≤ 1\): **None**
- Range: \(y > 2\): **Exponential graph**
- Range: \(y ≥ −4\): **Linear graph, Quadratic graph, Exponential graph, Absolute value graph**
- y-intercept: (0, 2): **Linear graph**
- y-intercept: (0, 3): **Exponential graph**
If you need any further clarifications or analyses, let me know!
1. **Linear graph with y-intercept at (0, 2) and another point at (1, 5)**:
- Domain: \(−∞ ≤ x ≤ ∞\) - **True**
- Range: \(−∞ ≤ y ≤ ∞\) - **True**
- Range: \(y ≤ 1\) - **False**
- Range: \(y > 2\) - **False**
- Range: \(y ≥ −4\) - **True**
- y-intercept: (0, 2) - **True**
- y-intercept: (0, 3) - **False**
2. **Quadratic graph with vertex at (4, -4) and x-intercepts at (2, 0) and (6, 0)**:
- Domain: \(−∞ ≤ x ≤ ∞\) - **True**
- Range: \(−∞ ≤ y ≤ ∞\) - **True**
- Range: \(y ≤ 1\) - **False**
- Range: \(y > 2\) - **False**
- Range: \(y ≥ −4\) - **True**
- y-intercept: (0, 2) - **False**
- y-intercept: (0, 3) - **False**
3. **Exponential graph with y-intercept at 3, and asymptote at y = 2**:
- Domain: \(−∞ ≤ x ≤ ∞\) - **True**
- Range: \(−∞ ≤ y ≤ ∞\) - **False** (Exponential graphs typically do not have an infinite range in the negative direction)
- Range: \(y ≤ 1\) - **False**
- Range: \(y > 2\) - **True** (since the asymptote at y = 2 means the graph approaches this value but does not cross it)
- Range: \(y ≥ −4\) - **True**
- y-intercept: (0, 2) - **False**
- y-intercept: (0, 3) - **True**
4. **Absolute value graph with vertex at (3, 1) and x-intercepts at (2, 0) and (4, 0)**:
- Domain: \(−∞ ≤ x ≤ ∞\) - **True**
- Range: \(−∞ ≤ y ≤ ∞\) - **False** (absolute value graphs typically have a minimum value)
- Range: \(y ≤ 1\) - **False**
- Range: \(y > 2\) - **False**
- Range: \(y ≥ −4\) - **True**
- y-intercept: (0, 2) - **False**
- y-intercept: (0, 3) - **False**
### Summary of findings:
- **Linear graph**: True for the domain, both ranges, and the y-intercept at (0, 2).
- **Quadratic graph**: True for the domain, both ranges (including y ≥ −4).
- **Exponential graph**: True for the domain, the range \(y > 2\), the range \(y ≥ −4\), and the y-intercept at (0, 3).
- **Absolute value graph**: True for the domain and the range \(y ≥ −4\).
### The graphs for which each statement is true:
- Domain: **All graphs**
- Range: \(−∞ ≤ y ≤ ∞\): **Linear graph, Quadratic graph**
- Range: \(y ≤ 1\): **None**
- Range: \(y > 2\): **Exponential graph**
- Range: \(y ≥ −4\): **Linear graph, Quadratic graph, Exponential graph, Absolute value graph**
- y-intercept: (0, 2): **Linear graph**
- y-intercept: (0, 3): **Exponential graph**
If you need any further clarifications or analyses, let me know!