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Chapter 2: Problem 48

Graphing Transformations Sketch the graph of the function, not by plotting points, but by starting with the graph of a standard function and applying transformations. $$y=2-\sqrt{x+1}$$

Short Answer

Expert verified
Start with \( y = \sqrt{x} \), shift left 1 unit, reflect over the x-axis, then shift up 2 units to graph \( y = 2 - \sqrt{x+1} \).

Step by step solution

01

Identify the Base Function

The base function here is the square root function, which is denoted as \( y = \sqrt{x} \). This is a standard function.
02

Apply the Horizontal Shift

The function \( y=\sqrt{x+1} \) indicates a horizontal shift to the left by 1 unit. This means the graph of \( y = \sqrt{x} \) will be moved 1 unit to the left.
03

Apply the Vertical Reflection

The function becomes \( y = -\sqrt{x+1} \), which indicates a reflection across the x-axis. This means we take the graph from Step 2 and reflect it downwards.
04

Apply the Vertical Shift

Finally, add 2 to the function to get \( y = 2 - \sqrt{x+1} \). This results in a vertical shift upwards by 2 units for the graph from Step 3.
05

Sketch the Graph

Start with the base graph of \( y = \sqrt{x} \), shift it left by 1 unit, reflect it across the x-axis, and then shift it upward by 2 units to get the final graph. Plotting the transformations collectively will give the desired graph of \( y=2-\sqrt{x+1} \).

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Key Concepts

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

Base Function
The base function plays a crucial role in understanding graph transformations. It serves as the starting point from which all transformations will be applied. In our case, the base function is the square root function, denoted as \( y = \sqrt{x} \). This function resembles a curve that begins at the origin and gradually increases. Understanding the shape and position of the base function is essential because each transformation is applied relative to the graph of this base function.聽
  • The base function \( y = \sqrt{x} \) has its graph starting at the point \((0, 0)\).
  • It increases steadily, staying in the first quadrant.
  • Any transformation applied will modify this initial graph.
Knowing the base function allows us to predict how the graphs will change as we apply various transformations.
Horizontal Shift
When working with graph transformations, a horizontal shift changes the position of the graph along the x-axis. This is done by adding or subtracting a number inside the function. In our example, the equation \( y = \sqrt{x+1} \) reflects a horizontal shift. This shift moves the graph to the left by 1 unit.
  • Typically, \( y = \sqrt{x} \) has its graph starting at the origin \((0, 0)\).
  • With \( y = \sqrt{x+1} \), the graph shifts left to start at \((-1, 0)\).
  • The same shape and pattern of the base function apply but at a new starting point.
Horizontal shifts help us reposition the function along the x-axis without altering its inherent shape.
Vertical Reflection
A vertical reflection changes the graph's orientation across the x-axis. It can be easily identified by a negative sign in front of the function. In this scenario, the transformation from \( y = \sqrt{x+1} \) to \( y = -\sqrt{x+1} \) indicates a vertical reflection.
  • Before reflection: The graph extends upwards to the right.
  • After reflection: The entire graph flips over the x-axis.
  • This results in a graph that extends downwards from the start point.
This reversal changes the direction of increase or decrease in the graph, but retains the original distances and angles seen in the base function.
Vertical Shift
The final transformation applied is a vertical shift, indicated by adding or subtracting a number at the end of the function. In our equation, the transformation from \( y = -\sqrt{x+1} \) to \( y = 2 - \sqrt{x+1} \) results in a vertical shift upwards by 2 units.
  • This shift moves every point on the graph up by 2 units.
  • The reflected graph, which previously touched the x-axis, is raised.
  • This action maintains the graph's shape and orientation.
The vertical shift alters the starting position along the y-axis without affecting the slope or curvature. It completes the graph transformation by positioning the graph at its intended final location.

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Most popular questions from this chapter

As a weather balloon is inflated, the thickness \(T\) of its rubber skin is related to the radius of the balloon by $$T(r)=\frac{0.5}{r^{2}}$$ where \(T\) and \(r\) are measured in centimeters. Graph the function \(T\) for values of \(r\) between 10 and \(100 .\)

DISCUSS DISCOVER: Minimizing a Distance When we seek a minimum or maximum value of a function, it is sometimes easier to work with a simpler function instead. (a) Suppose $$g(x)=\sqrt{f(x)}$$ where \(f(x) \geq 0\) for all \(x .\) Explain why the local minima and maxima of \(f\) and \(g\) occur at the same values of \(x .\) (b) Let \(g(x)\) be the distance between the point \((3,0)\) and the point \(\left(x, x^{2}\right)\) on the graph of the parabola \(y=x^{2}\) Express \(g\) as a function of \(x\) (c) Find the minimum value of the function \(g\) that you found in part (b). Use the principle described in part (a) to simplify your work.

DISCUSS: Solving an Equation for an Unknown Function In Exercises \(69-72\) of Section 2.7 you were asked to solve equations in which the unknowns are functions. Now that we know about inverses and the identity function (see Exercise 104 ), we can use algebra to solve such equations. For instance, to solve \(f \circ g=h\) for the unknown function \(f,\) we perform the following steps: \(f^{\circ} g=h \quad\) Problem: Solve for \(f\) \(f^{\circ} g^{\circ} g^{-1}=h^{\circ} g^{-1} \quad\) Compose with \(g^{-1}\) on the right \(f \circ I=h \cdot g^{-1} \quad \text { Because } g^{\circ} g^{-1}=I\) \(f=h e g^{-1} \quad\) Because \(f \circ I=f\) So the solution is \(f=h \circ g^{-1} .\) Use this technique to solve the equation \(f \circ g=h\) for the indicated unknown function. (a) Solve for \(f,\) where \(g(x)=2 x+1\) and \(h(x)=4 x^{2}+4 x+7.\) (b) Solve for \(q,\) where \(f(x)=3 x+5\) and \(h(x)=3 x^{2}+3 x+2.\)

A family of functions is given. In parts (a) and (b) graph all the given members of the family in the viewing rectangle indicated. In part (c) state the conclusions that you can make from your graphs. \(f(x)=x^{2}+c\) (a) \(c=0,2,4,6 ; \quad[-5,5]\) by \([-10,10]\) (b) \(c=0,-2,-4,-6 ;[-5,5]\) by \([-10,10]\) (c) How does the value of \(c\) affect the graph?

A linear function is given. (a) Find the average rate of change of the function between \(x=a\) and \(x=a+h .\) (b) Show that the average rate of change is the same as the slope of the line. $$g(x)=-4 x+2$$
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