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Magazine Chapter 1: Problem 19
Draw rough graphs of \(y=\sqrt{x}\) and \(y=\sqrt{x-4}\) and \(y=\sqrt{x}-4\). They are "half-parabolas" with infinite slope at the start.
Short Answer
Expert verified
Translate \( y=\sqrt{x} \) right for \( y=\sqrt{x-4} \) and shift down for \( y=\sqrt{x}-4 \).
Step by step solution
01
Understand the Function Shift
The graph of the function \( y = \sqrt{x} \) starts at the origin (0,0) and moves upwards to the right, forming a half-parabola shape. We will consider the transformations applied to it for \( y = \sqrt{x-4} \) and \( y = \sqrt{x} - 4 \).
02
Graph the Function y = sqrt(x)
Begin by sketching the graph of \( y = \sqrt{x} \). This graph is a curve that starts at the origin (0,0) and rises to the right, increasing at a decreasing rate. It is important to note that it's undefined for \( x < 0 \).
03
Graph y = sqrt(x-4)
This function represents a horizontal shift of \( y = \sqrt{x} \) to the right by 4 units. So, start graphing it at the point (4,0) instead of the origin and follow the same half-parabolic shape moving to the right.
04
Graph y = sqrt(x) - 4
For this function, we have a vertical shift of the graph of \( y = \sqrt{x} \) downward by 4 units. This means that the curve still starts at the origin, but lower across all points by 4 units. Essentially, the point (0,0) becomes (0,-4).
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Key Concepts
These are the key concepts you need to understand to accurately answer the question.
Function Shifts
In graph transformations, function shifts play a key role in altering the position of a graph. Understanding how function shifts work aids in accurately sketching and analyzing graphs.
- Horizontal Shift: When a function is shifted horizontally, each point on the graph moves left or right by a specific number of units. For example, in the function \( y = \sqrt{x-4} \), the graph of \( y = \sqrt{x} \) shifts 4 units to the right. This is because the inside of the square root is \( (x-4) \), shifting the graph to the right so that it starts at the point (4,0) instead of (0,0).
- Vertical Shift: A vertical shift involves moving the graph up or down. In our example, \( y = \sqrt{x} - 4 \), demonstrates a downward shift by 4 units. It affects the entire graph, changing points like (0,0) to (0,-4).
Half-Parabolas
Half-parabolas describe the shape of graphs stemming from square root functions. Unlike traditional parabolas, they only extend in one direction 鈥� forming a curve that starts at a point and rises or falls as you move along the x-axis.
- Characteristics: For \( y = \sqrt{x} \), the graph begins at (0,0) because \( \sqrt{0} = 0 \). The graph extends to the right, increasing at a slowing rate as \( x \) becomes larger. This creates a smooth curve known as a half-parabola.
- Slope: A key feature is the infinite slope at the start. This means that as \( x \) becomes smaller and approaches 0 from the right, the rate of increase is steep and becomes vertical at (0,0), though not defined for negative x values.
Graphing Techniques
Graphing techniques are crucial in creating accurate representations of functions on coordinate planes. When dealing with transformations like shifts, becoming familiar with graphing techniques simplifies the process.
- Starting Point: Identify where your graph begins. For basic \( y = \sqrt{x} \), it starts at (0,0). With transformations, note new starting points like (4,0) for \( y = \sqrt{x-4} \).
- Shape Consistency: While the position changes, the graph's shape remains consistent. Knowing this helps to maintain the curve's integrity as a half-parabola.
- Increment Plotting: Plot points at incremental x-values to guide the shape of the curve. With square root functions, using perfect square x-values like 1, 4, 9, simplifies calculations \( (\sqrt{1} = 1), (\sqrt{4} = 2), (\sqrt{9} = 3) \), assisting in plotting.
Square Root Functions
Square root functions are instrumental in various mathematical and real-life applications. They are rooted in taking the square root of the variable, representing a relationship that forms a unique graph.
- Basic Form: The simplest square root function is \( y = \sqrt{x} \). Its defining feature is to only output non-negative values for non-negative inputs, since square roots are defined as such. This results in the graph existing only in the first quadrant of the coordinate plane.
- Transformation Effects: Applying shifts or other transformations results in a visually different graph but maintains the fundamental structure of what a square root relationship entails.
- Applications: Beyond graphing, square root functions can describe phenomena in physics (e.g., calculating wave frequencies), biology (e.g., population models), and economics (e.g., calculating demand elasticity).
