Problem 40 - Precalculus

Question

Graph the exponential function using transformations. State the \(y\) -intercept, two additional points, the domain, the range, and the horizontal asymptote. $$f(x)=5^{x}-2$$

Step-by-Step Solution

Answer

The graph of \(f(x) = 5^x - 2\) has a y-intercept at (0, -1), additional points at (1, 3) and (2, 23). Its domain is \((-\infty, \infty)\), range is \((-2, \infty)\) and the horizontal asymptote is \(y = -2\).

Step 1: Identify the Parent Function

The parent function for the given function is \( f(x) = 5^x \). This is a basic exponential function where the base is 5.

Step 2: Determine the Transformations

The function \( f(x) = 5^x - 2 \) indicates a vertical shift. The term \(-2\) is a vertical shift downwards by 2 units from the parent function \( f(x) = 5^x \).

Step 3: Find the y-intercept

To find the \(y\)-intercept, set \(x = 0\) in \( f(x) = 5^x - 2 \). \[ f(0) = 5^0 - 2 = 1 - 2 = -1 \]. So, the \(y\)-intercept is \((0, -1)\).

Step 4: Find Two Additional Points

Choose two values for \(x\), such as 1 and 2:For \(x = 1\): \[ f(1) = 5^1 - 2 = 5 - 2 = 3 \]For \(x = 2\): \[ f(2) = 5^2 - 2 = 25 - 2 = 23 \]Thus, two additional points are \((1, 3)\) and \((2, 23)\).

Step 5: State the Domain and Range

The domain of an exponential function is all real numbers, written as \((-\infty, \infty)\). The range is affected by the vertical shift; since the graph is shifted down by 2 units, the range is \((-2, \infty)\).

Step 6: Identify the Horizontal Asymptote

The horizontal asymptote of the parent function \( 5^x \) is \(y = 0\). The vertical shift down by 2 units alters this to \(y = -2\).

Key Concepts

Transformations

In the world of exponential functions, transformations can help us visually understand how a function behaves as it changes. Transformations include shifts, stretches, and reflections.

• Vertical Shifts: The function \( f(x) = 5^x - 2 \) represents a vertical shift of the standard function \(f(x) = 5^x\). Here, subtracting 2 signifies that every point on the graph is moved 2 units downward.
• Horizontal Shifts: In our case, there are no horizontal shifts because no constant is added or subtracted inside the exponential function with the variable \(x\).
• Reflections and Stretches: This particular function doesn't involve a reflection or a stretch, but these would involve multiplying \(f(x)\) by a negative or a scalar factor respectively.

Understanding these transformations allows us to predict the graph's new position just by inspecting the equation.

Domain and Range

The domain of an exponential function like \( f(x) = 5^x - 2 \) is the set of all possible inputs \(x\). Exponential functions are defined for all real numbers, hence the domain is \((-fty, fty)\).

The range, however, is affected by transformations. In the parent function \(5^x\), the range is \((0, fty)\) as it grows exponentially and never reaches zero. However, the transformation \(-2\) shifts the entire graph down by 2 units, altering the range to \((-2, fty)\). This means any output value \(f(x)\) will be greater than -2.

Horizontal Asymptote

A horizontal asymptote is a line that the graph of a function approaches but never touches as \(x\) moves towards infinity or negative infinity.

For the parent function \(f(x) = 5^x\), the horizontal asymptote is \(y = 0\). This represents that as \(x\) becomes very large, \(5^x\) grows but never quite reaches 0. When we apply the transformation \(-2\) in \(f(x) = 5^x - 2\), the asymptote shifts downward, and now lies at \(y = -2\). While the function values approach \(-2\) as \(x\) decreases towards negative infinity, it will never equal -2, just getting closer and closer.

Graphing Exponential Functions

Graphing exponential functions like \(f(x) = 5^x - 2\) involves understanding their shape and behavior. Exponential growth is characterized by rapid increase at continuously faster rates.

When you graph \(f(x) = 5^x - 2\):

• Start by plotting key points like the \(y\)-intercept \((0, -1)\).
• Add points like \((1, 3)\) and \((2, 23)\) to determine the curve's steepness.
• Recall our transformations, ensuring the entire graph is shifted down by 2 units.

The graph displays a curve starting near \(-2\) and shooting upwards as \(x\) increases. Always remember that exponential functions will rise steeply, and the curve will never quite touch the horizontal asymptote, adding a smooth curve approaching this boundary from above.