Question
A linear function is expressed by a graph of a line that crosses both axes at the origin. An equation is written in slope-intercept form to express the same function as shown in the graph. From just this information, what statements can be made?
Choose the true statements.
The slope of the equation must be positive.
The y-intercept of the equation is zero.
It is possible that the line on the graph is horizontal.
The slope of the equation must be negative.
Choose the true statements.
The slope of the equation must be positive.
The y-intercept of the equation is zero.
It is possible that the line on the graph is horizontal.
The slope of the equation must be negative.
Answers
GPT-4o mini
Given that the linear function is represented by a graph of a line that crosses both axes at the origin, we can analyze the statements one by one:
1. **The slope of the equation must be positive.**
- This statement is **not necessarily true**. The slope can be positive, negative, or zero, depending on the angle of the line. Since it crosses at the origin, it can have any slope.
2. **The y-intercept of the equation is zero.**
- This statement is **true**. Since the line crosses the origin (0,0), the y-intercept (where the line crosses the y-axis) is indeed zero.
3. **It is possible that the line on the graph is horizontal.**
- This statement is **true**. A horizontal line at the origin (y = 0) has a slope of 0, which is a specific case of a linear function that also crosses the origin.
4. **The slope of the equation must be negative.**
- This statement is **not necessarily true**. As mentioned earlier, the slope can be positive, negative, or zero.
**To summarize the true statements:**
- The y-intercept of the equation is zero.
- It is possible that the line on the graph is horizontal.
1. **The slope of the equation must be positive.**
- This statement is **not necessarily true**. The slope can be positive, negative, or zero, depending on the angle of the line. Since it crosses at the origin, it can have any slope.
2. **The y-intercept of the equation is zero.**
- This statement is **true**. Since the line crosses the origin (0,0), the y-intercept (where the line crosses the y-axis) is indeed zero.
3. **It is possible that the line on the graph is horizontal.**
- This statement is **true**. A horizontal line at the origin (y = 0) has a slope of 0, which is a specific case of a linear function that also crosses the origin.
4. **The slope of the equation must be negative.**
- This statement is **not necessarily true**. As mentioned earlier, the slope can be positive, negative, or zero.
**To summarize the true statements:**
- The y-intercept of the equation is zero.
- It is possible that the line on the graph is horizontal.