Problem 32 - Intermediate Algebra

Question

Graph each of the functions. $$f(x)=\frac{3}{x+3}-4$$

Step-by-Step Solution

Answer

Vertical asymptote at \( x = -3 \), horizontal asymptote at \( y = -4 \), y-intercept at \((0, -3)\).

Step 1: Identify the Type of Function

The function given is \( f(x) = \frac{3}{x+3} - 4 \). This is a rational function, specifically a transformation of the parent function \( g(x) = \frac{1}{x} \). The transformations include a horizontal shift left by 3 units, vertical shift down by 4 units, and a vertical stretch by a factor of 3.

Step 2: Determine Vertical Asymptote

The horizontal shift affects the location of the vertical asymptote. The parent function \( g(x) = \frac{3}{x} \) has a vertical asymptote at \( x = 0 \). The function \( f(x) = \frac{3}{x+3} - 4 \) shifts the asymptote to \( x = -3 \) due to the term \((x+3)\).

Step 3: Determine Horizontal Asymptote

The horizontal asymptote is determined by the part \( -4 \) of the function \( f(x) = \frac{3}{x+3} - 4 \). This shifts the horizontal asymptote from \( y = 0 \) (of the parent function) to \( y = -4 \).

Step 4: Find Intercepts

To find the y-intercept, set \( x = 0 \): \[f(0) = \frac{3}{0+3} - 4 = \frac{3}{3} - 4 = 1 - 4 = -3.\]Thus, the y-intercept is \((0, -3)\). There is no x-intercept since the numerator 3 never equals zero.

Step 5: Sketch the Graph

Plot the vertical and horizontal asymptotes at \( x = -3 \) and \( y = -4 \) respectively. Mark the y-intercept at \((0, -3)\). Draw the curve approaching asymptotes in both directions, making sure the graph lies in the correct quadrants. Increasing or decreasing behavior is dictated by the sign and magnitude of the numerator.

Key Concepts

Graphing Functions

Graphing functions helps us visualize and understand the behavior of different types of functions on a coordinate plane. For the rational function \( f(x) = \frac{3}{x+3} - 4 \), graphing it involves recognizing how it transforms from its parent function \( g(x) = \frac{1}{x} \). When graphing functions, keep the following in mind:

• Identify transformations, such as shifts, stretches, or compressions, from the parent function.
• Locate important features, like asymptotes and intercepts, that impact the graph shape.

By understanding these characteristics, we can sketch the function's graph by plotting its asymptotes, and intercepts, and tracing the function's direction as it approaches the asymptotes.

This visual representation is crucial because it highlights how the function behaves. For \( f(x) = \frac{3}{x+3} - 4 \), we observe a horizontal shift left and a vertical shift down, helping us sketch the graph step by step. Making sure the graph approaches these lines without crossing them gives us the correct shape and position of the curve.

Vertical and Horizontal Asymptotes

In rational functions, vertical and horizontal asymptotes guide the graph's behavior, showing where the curve doesn't touch or cross.

• **Vertical asymptotes** occur where the denominator becomes zero. For \( f(x) = \frac{3}{x+3} - 4 \), the vertical asymptote is at \( x = -3 \), determined by setting the denominator \( (x+3) \) to zero.
• **Horizontal asymptotes** represent the function's end behavior as \( x \) approaches infinity or negative infinity. In \( f(x) = \frac{3}{x+3} - 4 \), the horizontal asymptote is at \( y = -4 \), due to the constant \( -4 \). This shifts from the parent function's \( y = 0 \) asymptote.

Finding these asymptotes is crucial because they frame where the graph approaches but never crosses. Asymptotes are the invisible boundaries of rational functions, showing us the natural limits of how these functions perform.
Being mindful of asymptotes when drawing a graph ensures that the behavior of the function is accurately depicted. Each asymptote shows us specific trends, letting us trace the curve as it stretches or shifts based on transformations.

Function Transformations

Function transformations modify the shape and position of the graph compared to its parent function. Transformations in \( f(x) = \frac{3}{x+3} - 4 \) include:

• A **horizontal shift** left by 3 units, as seen in the expression \( x+3 \), moving the graph to the left.
• A **vertical shift** down of 4 units, due to the \( -4 \), lowering the entire graph.
• A **vertical stretch** by a factor of 3, caused by multiplying the parent function by 3, making the graph steeper.

These transformations systematically alter how the graph behaves.

Understanding each type of transformation helps us predict what the graph will look like without having to plot numerous points manually. Recognizing these shifts and stretches allows us to take a simple function and tailor it to fit specific scenarios. The process of transformations creates a powerful tool for graphing diverse equations, fostering a deeper comprehension of mathematical relationships.