To draw the graph of the function [tex]\( f(x) = 5^{x-3} \)[/tex], follow these steps:
### Step 1: Understand the Function
The function [tex]\( f(x) = 5^{x-3} \)[/tex] is an exponential function where the base is 5 and the exponent is [tex]\( x-3 \)[/tex]. This indicates that the graph will have exponential growth characteristics, with a horizontal shift.
### Step 2: Identify Key Points
Calculate the value of the function at several key points:
- For [tex]\( x = 0 \)[/tex]:
[tex]\[
f(0) = 5^{0-3} = 5^{-3} = \frac{1}{125}
\][/tex]
- For [tex]\( x = 3 \)[/tex]:
[tex]\[
f(3) = 5^{3-3} = 5^0 = 1
\][/tex]
- For [tex]\( x = 6 \)[/tex]:
[tex]\[
f(6) = 5^{6-3} = 5^3 = 125
\][/tex]
### Step 3: Behavior at Infinity and Negative Infinity
- As [tex]\( x \to \infty \)[/tex], [tex]\( f(x) = 5^{x-3} \to \infty \)[/tex].
- As [tex]\( x \to -\infty \)[/tex], [tex]\( f(x) = 5^{x-3} \to 0 \)[/tex] because the exponent [tex]\( x-3 \)[/tex] becomes a large negative number.
### Step 4: Plotting Points
Consider plotting these points on the Cartesian plane:
- [tex]\( (0, \frac{1}{125}) \)[/tex]
- [tex]\( (3, 1) \)[/tex]
- [tex]\( (6, 125) \)[/tex]
### Step 5: Sketch the Curve
Based on these key points and behavior, sketch the curve:
1. Start by marking a few key points we computed on the graph.
2. Since [tex]\( f(x) = 5^{x-3} \)[/tex] is an increasing exponential function:
- The curve will pass through the points calculated.
- The value starts very close to 0 for negative [tex]\( x \)[/tex] and grows rapidly as [tex]\( x \)[/tex] increases.
3. The horizontal asymptote is [tex]\( y = 0 \)[/tex], indicating that as [tex]\( x \)[/tex] decreases towards negative infinity, the values of [tex]\( f(x) \)[/tex] approach 0 but never actually reach it.
### Step 6: Additional Points (If Needed)
For further accuracy, more points could be evaluated. For example:
- For [tex]\( x = 1 \)[/tex]:
[tex]\[
f(1) = 5^{1-3} = 5^{-2} = \frac{1}{25}
\][/tex]
- For [tex]\( x = 2 \)[/tex]:
[tex]\[
f(2) = 5^{2-3} = 5^{-1} = \frac{1}{5}
\][/tex]
- For [tex]\( x = 4 \)[/tex]:
[tex]\[
f(4) = 5^{4-3} = 5^1 = 5
\][/tex]
### Final Sketch
On your graph paper or graphing tool:
- The x-axis should range from, say, -1 to 10 to capture the behavior as [tex]\( x \)[/tex] grows.
- The y-axis should range from 0 to a suitably large value, such as 130 (to show [tex]\( f(x) \)[/tex] growing).
Plot the points:
- For negative [tex]\( x \)[/tex], [tex]\( f(x) \to 0 \)[/tex].
- [tex]\( (0, \frac{1}{125}) \)[/tex]
- [tex]\( (1, \frac{1}{25}) \)[/tex]
- [tex]\( (2, \frac{1}{5}) \)[/tex]
- [tex]\( (3, 1) \)[/tex]
- [tex]\( (4, 5) \)[/tex]
- [tex]\( (6, 125) \)[/tex]
Draw a smooth continuous curve through these points, ensuring to reflect the exponential growth and horizontal asymptotic behavior.
This completes the graph of [tex]\( f(x) = 5^{x-3} \)[/tex].
