Problem 92 - College Algebra Essentials

Question

Begin by graphing the square root function, \(f(x)=\sqrt{x} .\) Then use transformations of this graph to $$h(x)=2|x+3|$$

Step-by-Step Solution

Answer

The graph of the function \( h(x) = 2|x+3| \) originates from (-3,0) and forms a 'V' shape that is more steep than usual due to the scaling factor of 2.

Step 1: Graph the Square Root Function

Begin with sketching the graph for function \( f(x) = \sqrt{x} \). The square root function originates from (0,0) and curves upwards to the right for positive x values.

Step 2: Apply the Absolute Value Transformation

The function within the absolute value bars is \( x + 3 \) which represents a shift of the original function to the left by 3 places. Essentially, wherever 'x' used to be, it is now moved to \( x + 3 \). However, because of the absolute value, the output graph retains the magnitude but loses any negative indication, causing a mirror-image effect about the y-axis. The graph now originates from (-3,0) and forms a 'V' shape.

Step 3: Apply the Scaling Transformation

The factor of '2' on the outside of the absolute value bars scales the moment of the graph vertically, making it appear narrower. The slope of both arms of the 'V' shape now increase, resulting in the arms being more steep as the slope doubles with the scaling factor.

Key Concepts

Graphing Functions

Graphing functions is fundamental in understanding their behavior visually. It involves plotting the function’s output on a coordinate plane, allowing us to see how changes in the input affect the output. Let's break it down:

• **Domain and Range:** Understand which values of 'x' are permissible (domain) and what values 'f(x)' may take (range).
• **Shape and Position:** Identify the shape of the graph based on the function type (linear, quadratic, etc.) and its position related to axes.
• **Transformations:** These include shifts, reflections, and stretches or compressions, which modify the graph's position and shape.

To visualize a function effectively, start with identifying its basic shape using a simple equation, then make note of any transformations that will adjust this graph. Visual aids, like graphs, truly make abstract mathematical concepts more concrete and intuitive.

Square Root Function

The square root function is a classic example of a non-linear graph. It is represented mathematically as \( f(x) = \sqrt{x} \). Here's what you need to know:

• **Domain and Range:** The domain is all non-negative real numbers \( x \geq 0 \), as we can't have a square root of a negative number in real numbers. The range is also non-negative, \( f(x) \geq 0 \).
• **Graph Characteristics:** It starts at the origin (0,0) and increases gradually. The graph has a half-scrooping shape, curving upwards. The increase rate slows down as 'x' gets larger.
• **Transformations:** Graphs of square root functions can be shifted up/down or left/right using transformations. For example, \( f(x) = \sqrt{x} - 3 \) moves the graph 3 units downward.

This function serves as a building block for understanding more complex transformations, as it introduces concepts of translating graphs and dealing with roots.

Absolute Value Function

An absolute value function has a characteristic 'V' shape. The basic function \( f(x) = |x| \) can be transformed to obtain new graphs. Key points include:

• **Graph Characteristics:** The basic \( f(x) = |x| \) graph is symmetrical about the y-axis, forming a 'V' with its vertex at the origin.
• **Transformations:** Applying transformations inside the absolute value (like \( x + 3 \)) results in horizontal shifts. For example, \( f(x) = |x+3| \) shifts the vertex 3 units left.
  
  • **Scaling:** Multiplying by a constant, like 2 in \( f(x) = 2|x| \), stretches the graph vertically, making the arms of the 'V' steeper.
• **Effects:** Transformation can shift, reflect over axes, or scale the graph's steepness by changing slope coefficients.

Understanding these transformations on absolute value functions can help in grasping the effects on more complex functions, broadening the comprehension of function behavior.