Question

or Is this function linear, quadratic, or exponential?
x y
1 – 15
2 – 45
3 – 135
4 – 405
5 – 1,215

Answer 1

To determine whether the given function is linear, quadratic, or exponential, we can analyze the values of \( y \) given the corresponding values of \( x \).

Given values:
- When \( x = 1 \), \( y = -15 \)
- When \( x = 2 \), \( y = -45 \)
- When \( x = 3 \), \( y = -135 \)
- When \( x = 4 \), \( y = -405 \)
- When \( x = 5 \), \( y = -1215 \)

### Step 1: Check differences for linearity
First, let's check the first differences (subtract consecutive \( y \) values):

- \( -45 - (-15) = -30 \)
- \( -135 - (-45) = -90 \)
- \( -405 - (-135) = -270 \)
- \( -1215 - (-405) = -810 \)

The first differences are: \(-30, -90, -270, -810\). Since the first differences are not constant, the relation is not linear.

### Step 2: Check second differences for quadratic
Next, let's check the second differences (subtract consecutive first differences):

- \( -90 - (-30) = -60 \)
- \( -270 - (-90) = -180 \)
- \( -810 - (-270) = -540 \)

The second differences are: \(-60, -180, -540\). Since the second differences are not constant, the relation is not quadratic.

### Step 3: Check for exponential behavior
To check for an exponential pattern, we can look at the ratio of consecutive \( y \) values:

- \( \frac{-45}{-15} = 3 \)
- \( \frac{-135}{-45} = 3 \)
- \( \frac{-405}{-135} = 3 \)
- \( \frac{-1215}{-405} = 3 \)

The ratio of consecutive \( y \) values is constant at 3. This indicates that the function is exponential.

### Conclusion
The function is **exponential** and can be represented in the form \( y = a \cdot b^x \) where the base \( b \) is 3. In this case, the values for \( y \) can be expressed as:

\[
y = -15 \cdot 3^{x-1}
\]