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Chapter 1: Problem 51

Sketch the graph of the equation. $$ y=\sqrt{x}-4 $$

Short Answer

Expert verified
Sketch a downward-shifted square root graph starting at (0, -4).

Step by step solution

01

Identify the base function

The base function is the square root function, given by \( y = \sqrt{x} \). It is defined for \( x \geq 0 \) and has a basic graph that starts at the origin (0,0) and increases gradually to the right.
02

Analyze the transformation

The given function \( y = \sqrt{x} - 4 \) applies a vertical shift to the base function. The graph of \( y = \sqrt{x} \) is shifted 4 units downward. This means that each output value is reduced by 4.
03

Plot key points

Choose several x-values to calculate corresponding y-values for the transformed function. For example:- At \( x = 0 \), \( y = \sqrt{0} - 4 = -4 \).- At \( x = 1 \), \( y = \sqrt{1} - 4 = -3 \).- At \( x = 4 \), \( y = \sqrt{4} - 4 = -2 \).Plot these points on the coordinate grid.
04

Draw the graph

Connect the plotted points with a smooth curve that reflects the shape of the square root function shifted downward. Start from point (0, -4) and extend the curve to the right, maintaining the gradual upward slant characteristic of the square root function.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Square Root Function
The square root function is a fundamental mathematical function represented as \( y = \sqrt{x} \). This function is only defined for non-negative input values, meaning \( x \geq 0 \). The square root function produces a curve that starts from the origin, point (0, 0), and gradually increases to the right as x increases.
  • The shape of the curve reflects an ever-increasing, decelerating rate of change.
  • Its domain is all non-negative real numbers, and its range is also all non-negative real numbers.
A key characteristic of this function is its smooth, gentle slope. Visualizing this on a graph helps in understanding its growth over positive x-values.
Vertical Shift
Graph transformations include various types of modifications such as shifting, stretching, or reflecting. A vertical shift involves moving a graph up or down without altering its shape. The equation \( y = \sqrt{x} - 4 \) represents a vertical shift of the square root function.
  • The graph of \( y = \sqrt{x} \) is translated downward by 4 units.
  • This results in each point on the graph being 4 units lower than it would be in the base function.
This transformation affects only the y-coordinates of the graph, enabling students to visualize how functions can be systematically modified through simple operations like addition or subtraction.
Plotting Points
Plotting points is essential for accurately sketching any graph. In this equation \( y = \sqrt{x} - 4 \), starting by choosing specific x-values鈥攍ike 0, 1, and 4鈥攃an help illustrate the transformation:
  • At \( x = 0 \), \( y = \sqrt{0} - 4 = -4 \)
  • At \( x = 1 \), \( y = \sqrt{1} - 4 = -3 \)
  • At \( x = 4 \), \( y = \sqrt{4} - 4 = -2 \)
Once these points are computed, they can be plotted on a graph. Connecting these discrete points with a smooth curve will represent the continuous nature of the function graph.
Plotting points effectively creates a foundation from which the entire graph can be sketched accurately.
Coordinate Grid
When sketching a graph, utilizing a coordinate grid is crucial. The coordinate grid consists of two perpendicular lines鈥攖he x-axis (horizontal) and the y-axis (vertical)鈥攖hat form a plane. Each point on this plane is defined by a pair of values (x, y).
  • The origin, where both axes meet, is labeled as (0, 0).
  • Positive x-values lie to the right of the origin, and positive y-values lie above it.
The coordinate grid is like a map for your graph, offering a framework within which all points are accurately positioned and visualized.
Using the grid, a pattern arising from plotted points such as in the equation \( y = \sqrt{x} - 4 \) becomes immediately visible, aiding in understanding and practicing graph transformations.

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