Precalculus Mastering Precalculus Exponential Functions

Understanding Exponential Functions 🚀

An exponential function is a function of the form $f(x) = a^x$, where $a$ is a positive real number not equal to 1. The variable $x$ is the exponent.

Key Properties 🔑

• Base: $a > 0$ and $a ≠ 1$
• Domain: All real numbers $(-\infty, \infty)$
• Range: $(0, \infty)$ if $a > 0$
• Horizontal Asymptote: $y = 0$ (the x-axis)
• y-intercept: $(0, 1)$ because $a^0 = 1$

Graphing Exponential Functions 📈

To graph an exponential function, you can plot points or use transformations of the basic graph $y = a^x$.

Steps:

1. Create a table of values: Choose several x-values and compute the corresponding y-values.
2. Plot the points: Plot the points on a coordinate plane.
3. Draw the curve: Connect the points with a smooth curve.

Example: Graph $f(x) = 2^x$

import matplotlib.pyplot as plt
import numpy as np

x = np.linspace(-3, 3, 100)
y = 2**x

plt.plot(x, y)
plt.xlabel('x')
plt.ylabel('y')
plt.title('Graph of f(x) = 2^x')
plt.grid(True)
plt.show()

Transformations 🔄

Transformations can be applied to exponential functions to shift, stretch, or reflect the graph.

• Vertical Shift: $f(x) = a^x + k$ shifts the graph up ($k > 0$) or down ($k < 0$).
• Horizontal Shift: $f(x) = a^{x-h}$ shifts the graph right ($h > 0$) or left ($h < 0$).
• Vertical Stretch/Compression: $f(x) = c \cdot a^x$ stretches ($c > 1$) or compresses ($0 < c < 1$) the graph vertically.
• Reflection: $f(x) = -a^x$ reflects the graph across the x-axis.

Practice Problems ✍️

1. Graph $f(x) = 3^x - 2$
2. Graph $f(x) = (1/2)^x$
3. Solve for x: $4^x = 8$

Solutions:

1. $f(x) = 3^x - 2$: This is $3^x$ shifted down by 2 units.
2. $f(x) = (1/2)^x$: This is a decreasing exponential function.
3. $4^x = 8$: Rewrite as $(2^2)^x = 2^3$, so $2^{2x} = 2^3$. Thus, $2x = 3$ and $x = 3/2$.