Question

Describe the sequence of transformations from \(f(x)=\sqrt{x}\) to \(g\). Then sketch the graph of \(g\) by hand. Verify with a graphing utility. . \(g(x)=\sqrt{x+5}-2\)

Short answer

Function \(g(x)=\sqrt{x+5}-2\) is a transformation of \(f(x)=\sqrt{x}\) where it's shifted left by 5 units and down by 2 units. The corresponding graph would commence from the point (-5,-2) and extend to the right like a half parabola.

Step-by-step solution

Identify the transformations

First, observe the function \(g(x)=\sqrt{x+5}-2\). The +5 inside the square root indicates a horizontal shift to the left by 5 units. The -2 outside the square root indicates a vertical shift down by 2 units.

Apply the transformations to the graph of \(f(x)=\sqrt{x}\)

Draw the graph for \(f(x)=\sqrt{x}\), which starts from the origin (0,0) and extends towards right. Then, move each point on \(f(x)=\sqrt{x}\) left by 5 units and down by 2 units. This helps in obtaining the new function \(g(x)=\sqrt{x+5}-2\).

Sketch the graph

Plot the transformed function on a graph. The graph for \(g(x)=\sqrt{x+5}-2\) will commence from the point (-5,-2) instead of the origin, then extends towards right like a half parabola.

Verify with a graphing utility

This step involves double-checking the results. Use a graphing calculator or software to plug in \(g(x)=\sqrt{x+5}-2\) and confirm that the graph matches with the manually drawn sketch.

Key concepts

Horizontal Shift

A horizontal shift refers to moving a graph left or right on the coordinate plane. It's a common transformation seen in functions. When a function is expressed as \( \sqrt{x + c} \), the graph shifts horizontally. The sign of the constant \( c \) inside the function indicates the direction of the shift.

In our exercise, the function \( g(x) = \sqrt{x+5} \) shifts the original square root function to the left. The term \(+5\) indicates that every point on the graph of \( f(x) = \sqrt{x} \) moves 5 units left.

This shift doesn't change the vertical position of the graph. It simply relocates the entire shape along the x-axis.

Key points for horizontal shifts:

• The shift direction depends on the sign: addition shifts left, subtraction shifts right.
• It affects the input of the function, altering where you start plotting.

Vertical Shift

A vertical shift occurs when a graph moves up or down. In our function \( g(x) = \sqrt{x+5} - 2 \), the \(-2\) outside the square root signifies a vertical shift downward.

This transformation takes every point plotted from the horizontal shift and moves it down 2 units on the y-axis. The overall shape of the graph remains unchanged horizontally but shifts vertically.

Vertical shifts are easy to spot as they directly modify the function value. They're indicated by constants added or subtracted after computing the function's result.

Key aspects of vertical shifts:

• Positive constants shift the graph up, and negative constants shift it down.
• It affects the output, altering the vertical start of the plot.

Square Root Function

The square root function, denoted as \( f(x) = \sqrt{x} \), is a fundamental mathematical concept. It is part of the family of functions known as radical functions.

The graph of a basic square root function starts at the origin (0,0) and extends to the right. It's shaped like a gentle curve that initially inclines sharply before leveling out. This shape is sometimes described as half of a sideways parabola.

Characteristics of the square root function include:

• It only exists for non-negative inputs because negative numbers don't have real square roots.
• The function increases very slowly as \( x \) grows larger.
• It's only defined for \( x \geq 0 \).

Graphing Functions

Graphing functions means plotting points on a coordinate plane to represent mathematical expressions visually. For the function \( g(x) = \sqrt{x+5} - 2 \), you begin by identifying transformations from the square root function.

Start with the basic graph of \( f(x) = \sqrt{x} \). Apply the horizontal shift by moving all points 5 units to the left. Then, apply the vertical shift by moving them 2 units down.

When graphing:

• Begin with key points from the function like the starting point, usually computed from setting \( x \) to make the function simplest.
• Take note of how transformations affect these points and adjust their plot positions accordingly.
• Use software or graphing utilities to verify results, ensuring accuracy in transformation processes.