QQuestionMathematics
QuestionMathematics
How to graph an exponential function:
Identify the exponential function in the form y=a⋅b
x
.
Determine the y-intercept by evaluating the function at x= 0.
Find a few other points by choosing different x-values and solving for y.
Plot the points on a coordinate plane.
Draw a smooth curve through the points to represent the exponential function.
Note any asymptotes, typically the horizontal line y = 0 for basic exponential functions.
Label the graph appropriately.
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Answer
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Step 1:I'll solve the problem of graphing an exponential function step by step, following the specified formatting guidelines:
Step 2:: Identify the General Form of the Exponential Function
- $$x$$ is the independent variable
- a is the y-intercept (initial value)
Step 3:: Determine the Y-Intercept
- $$y = a \cdot b^{0} = a \cdot 1 = a
- Substitute x = 0 into the function
Step 4:: Calculate Additional Points
- When $$x = -1$$: $$y = 2 \cdot 3^{-1} = 2 \cdot \frac{1}{3} = \frac{2}{3}
- When x = 0: y = 2 \cdot 3^{0} = 2
Step 5:: Plot the Points
| -1 | $$\frac{2}{3}$$ |
Create a table of points: |-------|-------| | 0 | 2 | | 1 | 6 | | 2 | 18 |
Step 6:: Sketch the Graph
- Note the curve rises quickly as $$x$$ increases
- Start by plotting the points from the table - Connect the points with a smooth, curved line - The curve approaches but never touches the x-axis
Step 7:: Identify Key Characteristics
- Increasing function when $$b > 1
- Always positive y values - Continuous curve
Final Answer
The graph of y = 2 \cdot 3^{x} is an exponential curve that: - Passes through point (0, 2) - Rises rapidly for positive x - Approaches but never touches y = 0 - Has a smooth, increasing shape
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