Problem 42 - Intermediate Algebra

Question

Graph each of the functions. $$f(x)=-3 \sqrt{x-1}+2$$

Step-by-Step Solution

Answer

Graph the function by applying transformations to the square root function: shift right by 1, reflect and scale by 3, and shift up by 2.

Step 1: Understand the Function

The function given is \( f(x) = -3 \sqrt{x-1} + 2 \). This is a transformation of the basic square root function \( \sqrt{x} \). The transformations include a horizontal shift, vertical scaling, reflection, and a vertical shift.

Step 2: Horizontal Shift

The expression inside the square root, \( x - 1 \), indicates a horizontal shift to the right by 1 unit. Therefore, the function \( \sqrt{x} \) is shifted to become \( \sqrt{x-1} \).

Step 3: Vertical Scaling and Reflection

The coefficient \(-3\) outside the square root indicates a vertical scaling by a factor of 3 and also a reflection over the x-axis. This means that any value of the function will be tripled and negated. So, the function is \( -3 \sqrt{x-1} \).

Step 4: Vertical Shift

The constant \(+2\) is a vertical shift that moves the graph up by 2 units. This transformation changes the function to \( f(x) = -3 \sqrt{x-1} + 2 \).

Step 5: Determine the Domain

Since \( \sqrt{x-1} \) is only defined for \( x-1 \geq 0 \), the domain of \( f(x) \) is \( x \geq 1 \).

Step 6: Finding Key Points

To graph the function, find key points: - When \( x = 1 \), \( f(x) = -3 \sqrt{1-1} + 2 = 2 \).- When \( x = 2 \), \( f(x) = -3 \sqrt{2-1} + 2 = -1 \).- When \( x = 5 \), \( f(x) = -3 \sqrt{5-1} + 2 = -4 + 2 = -2 \).

Step 7: Sketch the Graph

Using the key points found in Step 6, plot these points on a graph, and draw a smooth curve starting at (1,2), passing through (2,-1), and continuing according to the function form for \( x \geq 1 \). The graph is a reflected and vertically stretched version of the square root function, shifted up by 2.

Key Concepts

Square Root Function

The square root function is one of the fundamental functions in mathematics. Its standard form is represented as \( \sqrt{x} \). This function produces a set of outputs that curves gently upwards, starting from the origin (0,0). The square root sign, \( \sqrt{} \), denotes that we are looking for a number that, when multiplied by itself, gives the number under the root. Therefore, \( \sqrt{x} \) corresponds to the positive root of \( x \).Key characteristics of the square root function include:

• Domain: Non-negative values, meaning \( x \geq 0 \).
• Range: Non-negative values, meaning \( f(x) \geq 0 \).
• Behavior: Increasing and concave down, indicating a gentle upward trend.

The square root function is significant as it builds the foundation for understanding more complex transformations applied to it, like in our exercise.

Function Transformation

Function transformation involves modifying the basic function to produce a graph that appears shifted, stretched, compressed, or flipped. For the function \( f(x) = -3 \sqrt{x-1} + 2 \), multiple transformations occur.- **Horizontal Shifting:** The term \( (x-1) \) inside the square root signifies a shift to the right by 1 unit. This means if the original root starts at 0, it now starts at 1.
- **Vertical Scaling and Reflection:** The coefficient \(-3\) modifies the function in two ways. It reflects the graph over the x-axis, causing it to flip upside down, and it vertically scales it by a factor of 3, making it steeper.
- **Vertical Shifting:** The term \(+2\) adds a vertical shift, moving the entire graph 2 units upwards.Understanding these transformations helps predict the graph's new shape and position.

Domain and Range

The domain and range of a function determine the set of possible input and output values, respectively. For the transformed function \( f(x) = -3 \sqrt{x-1} + 2 \), discerning these sets is crucial.- **Domain:** Refers to all allowable inputs. Since the square root function \( \sqrt{x-1} \) only accepts non-negative inputs, we set the inside \( x-1 \geq 0 \), giving us \( x \geq 1 \).
- **Range:** Describes the possible outputs. Due to the reflection and scaling, the output values will start from 2 (when \( x=1 \)) and decrease to negative infinity. Thus, the range is \( f(x) \leq 2 \).
Understanding the domain and range ensures proper graph plotting within the allowable values.

Graphing Techniques

Graphing functions involve plotting several key points to determine the shape and behavior of the function. For the function \( f(x) = -3 \sqrt{x-1} + 2 \), the following steps aid in effective graphing:1. **Identify Key Points:** Compute values at strategic input points, like \( x = 1, 2, 5 \). These points (1,2), (2,-1), (5,-2) establish the main curve.
2. **Plot and Connect:** Mark these points on the graph. Connect them smoothly, adhering to the understanding that after \( x = 1 \), the function decreases continually.
3. **Incorporate Transformations:** Reflect the graph downwards while ensuring it stays steeper and shifted correctly according to the transformations.These techniques help create a precise visual representation of the function, making the effect of each transformation apparent.