Explore Westonci.ca, the leading Q&A site where experts provide accurate and helpful answers to all your questions. Get quick and reliable answers to your questions from a dedicated community of professionals on our platform. Join our Q&A platform to connect with experts dedicated to providing accurate answers to your questions in various fields.
Sagot :
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].
### 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].
Thank you for your visit. We're dedicated to helping you find the information you need, whenever you need it. We appreciate your visit. Our platform is always here to offer accurate and reliable answers. Return anytime. Westonci.ca is here to provide the answers you seek. Return often for more expert solutions.