To graph the exponential function:
$g(x) = \left(\frac{1}{4}\right)^x - 2$ we analyze its properties:

Step 1: Identify Key Features

1. Base and Growth/Decay:

   • The base is $\frac{1}{4}$, which is a fraction between 0 and 1, meaning it represents exponential decay.

2. Horizontal Asymptote:

   • The function has a horizontal asymptote at $y = -2$ because of the vertical shift.

3. Intercepts:

   • Y-Intercept: Set $x = 0$: $g(0) = \left(\frac{1}{4}\right)^0 - 2 = 1 - 2 = -1$ So, the y-intercept is $(0, -1)$.

   • X-Intercept: Set $g(x) = 0$ and solve for $x$: $\left(\frac{1}{4}\right)^x - 2 = 0$ $\left(\frac{1}{4}\right)^x = 2$ Take the logarithm on both sides: $x \log\left(\frac{1}{4}\right) = \log(2)$ Since $\log(1/4) = -\log(4)$, we get: $x = \frac{\log(2)}{-\log(4)}$ $x = \frac{\log(2)}{-2\log(2)} = -\frac{1}{2}$ So, the x-intercept is $(-0.5, 0)$.

Step 2: Behavior of the Function

• As $x \to \infty$, $g(x) \to -2$.
• As $x \to -\infty$, $g(x) \to \infty$.
• The function is always decreasing.

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Let's generate the graph now. numpy as np import matplotlib.pyplot as plt

Define the function g(x) = (1/4)^x - 2

def g(x): return (1/4)**x - 2

Generate x values

x_values = np.linspace(-3, 3, 400) y_values = g(x_values)

Plot the function

plt.figure(figsize=(8, 5)) plt.plot(x_values, y_values, label=r'$g(x) = \left(\frac{1}{4}\right)^x - 2$', color='blue')

Plot the asymptote y = -2

plt.axhline(y=-2, color='red', linestyle='dashed', label='Asymptote: $y=-2$')

Plot intercepts

plt.scatter([0], [-1], color='green', label='Y-Intercept (0,-1)') plt.scatter([-0.5], [0], color='purple', label='X-Intercept (-0.5,0)')

Labels and title

plt.xlabel('x') plt.ylabel('g(x)') plt.title('Graph of $g(x) = (1/4)^x - 2$') plt.axhline(0, color='black', linewidth=1) plt.axvline(0, color='black', linewidth=1) plt.legend() plt.grid()

Show the plot

plt.show()Here is the graph of $g(x) = \left(\frac{1}{4}\right)^x - 2$:

• The red dashed line represents the horizontal asymptote at $y = -2$.
• The green point marks the y-intercept at $(0, -1)$.
• The purple point marks the x-intercept at $(-0.5, 0)$.
• The function decreases towards the asymptote as $x \to \infty$ and grows rapidly as $x \to -\infty$.

Would you like more details or another transformation explained?

Related Questions:

1. How does changing the base (e.g., $\frac{1}{2}$ instead of $\frac{1}{4}$) affect the graph?
2. What is the effect of shifting the function by adding or subtracting from $x$?
3. How would you determine the domain and range of this function?
4. What happens if we reflect this function over the x-axis?
5. Can you express this function in logarithmic form?

Tip:

In exponential decay functions like this, the rate of decay is determined by the base. The closer the base is to 0, the faster the function approaches the asymptote!