Question

Graph 1:
time(hours) temperature(Celsius)
3 5
6 0
9 -5
12 -10

Graph 2:
x y
1 1
2 8
3 27
4 64
5 125
6 216
7 ?

Equation 1:
25 = x to the second power + y to the second power

Equation 2:
(y+7) = 3 (x-2)
Find out if these problems are linear, functions, non linear, and/or not functions

Answer 1

Let's analyze each of the scenarios you've presented: Graphs 1 and 2, as well as Equations 1 and 2.

### Graph 1:
- **Data**:
- (3, 5)
- (6, 0)
- (9, -5)
- (12, -10)

#### Analysis:
1. **Is it linear?**: To determine if the relationship is linear, we can check if the differences in temperature change at a constant rate as time increases. Let's calculate the differences:
- From \(3\) to \(6\): \(0 - 5 = -5\)
- From \(6\) to \(9\): \(-5 - 0 = -5\)
- From \(9\) to \(12\): \(-10 - (-5) = -5\)

The change in temperature is the same (-5 degrees for each 3 hours). Hence, the relationship is linear.

2. **Is it a function?**: Yes, this is a function because for each value of time, there is exactly one corresponding temperature.

### Graph 2:
- **Data**:
- (1, 1)
- (2, 8)
- (3, 27)
- (4, 64)
- (5, 125)
- (6, 216)
- (7, ?)

#### Analysis:
1. **Is it linear?**: To determine if the relationship is linear, we can compute the differences of y-values and check if they are constant:
- Differences:
- From 1 to 8: \(8 - 1 = 7\)
- From 8 to 27: \(27 - 8 = 19\)
- From 27 to 64: \(64 - 27 = 37\)
- From 64 to 125: \(125 - 64 = 61\)
- From 125 to 216: \(216 - 125 = 91\)

Since these differences are not constant, the relationship is **not linear**.

2. **Is it a function?**: Yes, this is a function because for each x-value, there is exactly one corresponding y-value.

### Equation 1:
- **Equation**: \(25 = x^2 + y^2\)

#### Analysis:
1. **Is it linear?**: No, this is not linear because it is a quadratic equation (both \(x\) and \(y\) are raised to the power of 2).

2. **Is it a function?**: It can be a function in some contexts, but this equation describes a circle (specifically a circle of radius 5 centered at the origin). However, it does not pass the vertical line test because for some x-values, there are two corresponding y-values. Thus, it is **not a function**.

### Equation 2:
- **Equation**: \((y+7) = 3(x-2)\)

#### Analysis:
1. **Is it linear?**: Yes, this equation can be rearranged into the slope-intercept form (y = mx + b):
- Expanding gives: \(y + 7 = 3x - 6\)
- Thus, \(y = 3x - 13\).
- This is a linear equation.

2. **Is it a function?**: Yes, this is a function because for each x-value, there is exactly one corresponding y-value.

### Summary:
- **Graph 1**: Linear, Function
- **Graph 2**: Non-linear, Function
- **Equation 1**: Non-linear, Not a Function
- **Equation 2**: Linear, Function