Question

draw the graph of f(x) = 6^{x} - 1

Explanation:

Step1: Identify the parent function

The parent function is \( y = 6^x \), which is an exponential growth function with a horizontal asymptote at \( y = 0 \), passing through \( (0, 1) \).

Step2: Analyze the transformation

The function \( f(x) = 6^{x}-1 \) is a vertical shift of the parent function \( y = 6^x \) down by 1 unit. So the horizontal asymptote shifts from \( y = 0 \) to \( y = -1 \).

Step3: Find key points

• When \( x = 0 \): \( f(0) = 6^{0}-1 = 1 - 1 = 0 \), so the point is \( (0, 0) \).
• When \( x = 1 \): \( f(1) = 6^{1}-1 = 6 - 1 = 5 \), so the point is \( (1, 5) \).
• When \( x = -1 \): \( f(-1) = 6^{-1}-1=\frac{1}{6}-1 = -\frac{5}{6}\approx - 0.83 \), so the point is \( (-1, -\frac{5}{6}) \).

Step4: Sketch the graph

1. Draw the horizontal asymptote \( y = -1 \) (a dashed line).
2. Plot the points \( (0, 0) \), \( (1, 5) \), \( (-1, -\frac{5}{6}) \).
3. Draw a smooth curve through these points, approaching the asymptote \( y = -1 \) as \( x\to -\infty \) and increasing rapidly as \( x\to \infty \), similar to the shape of \( y = 6^x \) but shifted down by 1.

Answer:

To draw the graph of \( f(x)=6^{x}-1 \):

1. Draw the horizontal asymptote \( y = -1 \).
2. Plot points like \( (0, 0) \), \( (1, 5) \), \( (-1, -\frac{5}{6}) \).
3. Draw a smooth exponential curve through these points, approaching \( y = -1 \) as \( x\to -\infty \) and increasing as \( x\to \infty \).