Question

Henry earns $0.50 in interest from his savings account every year.
Equation:
f(x)=1.5x 5
f(x)=2x+100 4 = 7
f(x)=0.5x 3
f(x)=100x 2
f(x)=100(0.5)^x
f(x)=100(0.98)^x
f(x)=100(1.5)^x
f(x)=100(2)^x
Is this an example of a linear or exponential relationship?

Answer 1

To determine whether each function represents a linear or exponential relationship, we can analyze the form of each function.

1. **Linear relationships** have the form \( f(x) = mx + b \), where \( m \) is the slope (constant) and \( b \) is the y-intercept.
2. **Exponential relationships** have the form \( f(x) = a \cdot b^x \), where \( a \) is a constant, and \( b \) is the base of the exponent.

Now let's categorize the equations you've provided:

1. \( f(x) = 1.5x + 5 \) - **Linear** (Form: \( mx + b \))
2. \( f(x) = 2x + 100 \) - **Linear** (Form: \( mx + b \))
3. \( f(x) = 0.5x + 3 \) - **Linear** (Form: \( mx + b \))
4. \( f(x) = 100x + 2 \) - **Linear** (Form: \( mx + b \))
5. \( f(x) = 100(0.5)^x \) - **Exponential** (Form: \( a \cdot b^x \))
6. \( f(x) = 100(0.98)^x \) - **Exponential** (Form: \( a \cdot b^x \))
7. \( f(x) = 100(1.5)^x \) - **Exponential** (Form: \( a \cdot b^x \))
8. \( f(x) = 100(2)^x \) - **Exponential** (Form: \( a \cdot b^x \))

In summary:
- The first four equations represent **linear** relationships.
- The last four equations represent **exponential** relationships.