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Chapter 3: Problem 55

Use transformations to sketch a graph of \(f\). \(f(x)=\sqrt{x-3}+2\)

Short Answer

Expert verified
Shift the graph of \( y = \sqrt{x} \) right 3 units and up 2 units.

Step by step solution

01

Identify the Base Function

The base function in this exercise is the square root function, which is expressed as \( y = \sqrt{x} \). The graph of this function is known to be a curve that starts at the origin (0,0) and increases steadily in the first quadrant.
02

Horizontal Shift

The given function \( f(x) = \sqrt{x-3} + 2 \) shows a horizontal shift. By comparing it to \( y = \sqrt{x} \), notice the \( x-3 \) inside the square root. This indicates a rightward shift of the graph by 3 units.
03

Vertical Shift

Additionally, the function adds 2 outside of the square root: \( \sqrt{x-3} + 2 \). This results in shifting the graph of \( y = \sqrt{x-3} \) upwards by 2 units along the y-axis.
04

Sketch the Transformed Graph

Start by sketching the original \( y = \sqrt{x} \) graph. Then apply the transformations: shift this graph 3 units to the right and 2 units upwards. The new graph should start at the point (3,2) and follow the same shape as the original square root graph.

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Key Concepts

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

Graphing Functions
Graphing functions involves plotting the values of a function on a coordinate plane. It provides a visual representation of how the function behaves across different inputs. Let's consider the example of the square root function:
  • The basic form of the square root function is given by \( y = \sqrt{x} \).
  • This graph looks like a curve that starts at the origin (0,0).
  • It steadily rises in the first quadrant of the graph.
To graph a function more effectively, it's important to understand transformations. Two common transformations are horizontal and vertical shifts, which allow you to adjust the position of the graph without changing its shape.To start graphing any transformed version of a function:
  • Draw the basic graph of the original function.
  • Apply the necessary transformations like shifts.
  • Sketch the new graph based on these adjustments.
By using these techniques, you'll be able to see the impact of changes within the function's formula visually.
Horizontal Shift
A horizontal shift is a transformation that moves a graph left or right along the x-axis. In function notation, a horizontal shift is reflected by changes inside the function's argument. Consider \( f(x) = \sqrt{x-3} \) as our example:
  • Here, the expression inside the square root, \( x-3 \), indicates a horizontal shift.
  • The graph of the original base function, \( y = \sqrt{x} \), gets shifted right by 3 units to become \( y = \sqrt{x-3} \).
Visualize this shift by considering what happens to the start point of the graph:
  • In the base graph \( y = \sqrt{x} \), it starts at (0,0).
  • After shifting right by 3 units, the new starting point is (3,0).
  • The shape of the graph remains unchanged; only its position is altered.
Understanding horizontal shifts allows you to predict the movement of graphs just by looking at changes within the function's formula.
Vertical Shift
Vertical shifts move graphs up or down along the y-axis. These shifts are caused by adding or subtracting a constant outside the main part of a function. Let's explore this with the function \( f(x) = \sqrt{x-3} + 2 \):
  • The "+2" at the end of the function prompts a vertical shift.
  • This shifts the entire graph of \( y = \sqrt{x-3} \) up by 2 units.
To visualize a vertical shift:
  • Consider the starting point after the horizontal shift, which was (3,0).
  • Shifting up by 2 units places the starting point at (3,2).
  • Once again, the overall shape of the graph remains the same; it's merely moved upwards.
Vertical shifts, like horizontal ones, are straightforward to implement once you spot how constants are added or subtracted to the function. This understanding helps in quickly determining how a graph's position changes on a coordinate plane.

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Most popular questions from this chapter

Two functions, \(f\) and \(g,\) are related by the given equation. Use the numerical representation of \(f\) to make a numerical representation of \(\mathbf{g}\). \(g(x)=f(x-2)\) $$\begin{array}{rrrrrr}x & -4 & -2 & 0 & 2 & 4 \\\f(x) & 5 & 2 & -3 & -5 & -9\end{array}$$

The points \((-12,6),(0,8),\) and \((8,-4)\) lie on the graph of \(y=f(x)\). Determine three points that lie on the graph of \(y=g(x)\). \(g(x)=f(x)-3\)

Complete the following. (a) Write the equation as \(a x^{2}+b x+c=0\) with \(a>0\) (b) Calculate the discriminant \(b^{2}-4 a c\) and determine the number of real solutions. (c) Solve the equation. $$ 2 x^{2}+3 x=12-2 x $$

The stopping distance \(d\) in feet for a car traveling at \(x\) miles per hour is given by \(d(x)=\frac{1}{12} x^{2}+\frac{11}{9} x\). Determine the driving speeds that correspond to stopping distances between 300 and 500 feet, inclusive. Round speeds to the nearest mile per hour.

If the graph of \(y=f(x)\) undergoes a vertical stretch or shrink to become the graph of \(y=g(x),\) do these two graphs have the same \(x\) -intercepts? \(y\) -intercepts? Explain your answers.
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