Question
For each table, graph and equation, select whether it is linear and whether it is a function.
x 3,6,9,12
y 5,0,-5,-10
25=x²+y²
(y+7)=3(x-2)
x 1,2,3,4,5,6,7
y 1,8,27,64,125,216,?
x 3,6,9,12
y 5,0,-5,-10
25=x²+y²
(y+7)=3(x-2)
x 1,2,3,4,5,6,7
y 1,8,27,64,125,216,?
Answers
GPT-4o mini
Let's analyze each case one by one.
1. **Table:**
- **x:** 3, 6, 9, 12
- **y:** 5, 0, -5, -10
To check if it's a linear function, we look for a constant difference in y-values when x increases by a constant amount. The differences in y-values are:
- From 5 to 0: -5
- From 0 to -5: -5
- From -5 to -10: -5
Since the differences are constant, the relationship is linear.
**Is it a function?** Yes, because for each input (x value), there is exactly one output (y value).
**Conclusion:** Linear: Yes, Function: Yes
2. **Equation:** \( 25 = x^2 + y^2 \)
This equation represents a circle with radius 5 centered at the origin. Circles are not linear functions.
**Conclusion:** Linear: No, Function: No (because a vertical line would intersect the circle at multiple points)
3. **Equation:** \( (y + 7) = 3(x - 2) \)
This equation can be simplified to slope-intercept form (y = mx + b):
- First, distribute: \( y + 7 = 3x - 6 \)
- Then, isolate y: \( y = 3x - 13 \)
This is a linear equation.
**Is it a function?** Yes, because for each x value, there is exactly one y value.
**Conclusion:** Linear: Yes, Function: Yes
4. **Table:**
- **x:** 1, 2, 3, 4, 5, 6, 7
- **y:** 1, 8, 27, 64, 125, 216, ?
To find the pattern in y-values, we notice:
- \( 1^3 = 1 \)
- \( 2^3 = 8 \)
- \( 3^3 = 27 \)
- \( 4^3 = 64 \)
- \( 5^3 = 125 \)
- \( 6^3 = 216 \)
- \( 7^3 = 343 \)
The relationship between x and y is not linear (as it is cubic).
**Is it a function?** Yes, because for each x value, there is a unique y value.
**Conclusion:** Linear: No, Function: Yes
### Summary of Conclusions:
1. Table: Linear: Yes, Function: Yes
2. Equation (circle): Linear: No, Function: No
3. Equation (line): Linear: Yes, Function: Yes
4. Table (cubic function): Linear: No, Function: Yes
1. **Table:**
- **x:** 3, 6, 9, 12
- **y:** 5, 0, -5, -10
To check if it's a linear function, we look for a constant difference in y-values when x increases by a constant amount. The differences in y-values are:
- From 5 to 0: -5
- From 0 to -5: -5
- From -5 to -10: -5
Since the differences are constant, the relationship is linear.
**Is it a function?** Yes, because for each input (x value), there is exactly one output (y value).
**Conclusion:** Linear: Yes, Function: Yes
2. **Equation:** \( 25 = x^2 + y^2 \)
This equation represents a circle with radius 5 centered at the origin. Circles are not linear functions.
**Conclusion:** Linear: No, Function: No (because a vertical line would intersect the circle at multiple points)
3. **Equation:** \( (y + 7) = 3(x - 2) \)
This equation can be simplified to slope-intercept form (y = mx + b):
- First, distribute: \( y + 7 = 3x - 6 \)
- Then, isolate y: \( y = 3x - 13 \)
This is a linear equation.
**Is it a function?** Yes, because for each x value, there is exactly one y value.
**Conclusion:** Linear: Yes, Function: Yes
4. **Table:**
- **x:** 1, 2, 3, 4, 5, 6, 7
- **y:** 1, 8, 27, 64, 125, 216, ?
To find the pattern in y-values, we notice:
- \( 1^3 = 1 \)
- \( 2^3 = 8 \)
- \( 3^3 = 27 \)
- \( 4^3 = 64 \)
- \( 5^3 = 125 \)
- \( 6^3 = 216 \)
- \( 7^3 = 343 \)
The relationship between x and y is not linear (as it is cubic).
**Is it a function?** Yes, because for each x value, there is a unique y value.
**Conclusion:** Linear: No, Function: Yes
### Summary of Conclusions:
1. Table: Linear: Yes, Function: Yes
2. Equation (circle): Linear: No, Function: No
3. Equation (line): Linear: Yes, Function: Yes
4. Table (cubic function): Linear: No, Function: Yes