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Chapter 3: Problem 54

Use transformations to sketch a graph of \(f\). \(f(x)=\sqrt{x}+1\)

Short Answer

Expert verified
The graph of \\(f(x)\\) is \\(\sqrt{x}\\) shifted up by 1 unit.

Step by step solution

01

Identify Base Function

The given function is \(f(x) = \sqrt{x} + 1\). The base function for this is \(g(x) = \sqrt{x}\). This function is the square root function, and we know its basic shape: it starts at the origin (0,0) and increases gradually to the right.
02

Determine Transformations

The function \(f(x) = \sqrt{x} + 1\) can be rewritten as \(f(x) = \sqrt{x} + 1\). This transformation involves a vertical shift. Specifically, the graph of \(\sqrt{x}\) is shifted upward by 1 unit.
03

Apply Transformation

To apply the vertical shift, take each output \(y = \sqrt{x}\) of the base function and add 1. Hence, the transformed function will have points such that for any \(x\), \(y = \sqrt{x} + 1\). For example, the point (4, 2) on \(g(x)\) becomes (4, 3) on \(f(x)\).
04

Sketch the Graph

Draw the graph based on the transformed function. Start by plotting key points such as (0,1), (1,2), (4,3) based on our transformation. Continue by drawing a smooth curve that starts at (0,1) and increases as \(x\) increases, similar to the shape of the square root function but shifted upward by 1 unit.

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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 type of function in mathematics. It is represented by the form \(g(x) = \sqrt{x}\). At its core, the square root function produces a curve that begins at the origin, point (0,0), and gently rises to the right. This distinctive shape is due to the nature of the square root operation, which increases as the input, \(x\), becomes larger.
  • The square root function is defined for all non-negative numbers, meaning it only takes values for \(x \geq 0\).
  • It is crucial to understand that the output is always non-negative since square roots of negative numbers are not defined within the realm of real numbers.
  • Each input \(x\) in the function has a direct correspondence with an output \(y\) where \(y = \sqrt{x}\).
Understanding the square root function鈥檚 progression is essential for graph transformations, such as vertical shifts.
Graph Sketching
Graph sketching is a process that involves drawing a function鈥檚 graph based on its algebraic expression. It's essential in understanding how functions behave by giving a visual representation.
  • Identify the base function: Start by recognizing the simplest form of the function. In our case, it is \(g(x) = \sqrt{x}\).
  • Determine the transformations: Look for any changes from the base function, such as shifts, stretches, or reflections.
  • Apply the transformations: Implement these changes to the graph by adjusting key points.
  • Plot key points: Plotting a few critical points can help guide the curve accurately.
  • Draw the curve: Connect the points smoothly to represent the function accurately.
These steps make sketching a graph less daunting. They ensure that each transformation step is methodically applied, showcasing how the function changes visually.
Vertical Shift
A vertical shift in a graph involves moving the entire graph up or down along the y-axis. In the function \(f(x) = \sqrt{x} + 1\), a vertical shift occurs when 1 is added to the output of the square root function.
  • Function notation: The expression \(f(x) = \sqrt{x} + 1\) indicates a vertical shift of 1 unit upwards.
  • Effect on the graph: Every point on the base function \(g(x) = \sqrt{x}\) is shifted upward by 1 unit, changing from (\(x, y\)) to (\(x, y+1\)).
  • Example: A point such as (4,2) on the original square root graph will become (4,3) after the vertical shift.
Vertical shifts modify the range of the function but maintain the same shape, just translated vertically. This transformation does not affect the domain, allowing the new graph to reflect the same horizontal extent as the original.

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Most popular questions from this chapter

A baseball is dropped from a stadium seat that is 75 feet above the ground. Its height \(s\) in feet after \(t\) seconds is given by \(s(t)=75-16 t^{2} .\) Estimate to the nearest tenth of a second how long it takes for the baseball to strike the ground.

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 \(y=g(x)\). \(g(x)=f(x)+2\)

The table lists numbers of Wal-Mart employees \(E\) in millions, \(x\) years after 1987 . $$ \begin{array}{cccccc} x & 0 & 5 & 10 & 15 & 20 \\ \hline B & 0.20 & 0.38 & 0.68 & 1.4 & 2.2 \end{array} $$ (a) Evaluate \(E(15)\) and interpret the result. (b) Find a quadratic function \(f\) that models these data. (c) Graph the data and function \(f\) in the same \(x y\) -plane. (d) Use \(f\) to estimate the year when the number of cmployees may reach 3 million.

The table lists the velocity and distance raveled by a falling object for various elapsed times. $$ \begin{array}{|rcccccc|} \hline \text { Time (sec) } & 0 & 1 & 2 & 3 & 4 & 5 \\ \hline \text { Velocity (fl/sec) } & 0 & 32 & 64 & 96 & 128 & 160 \\ \hline \text { Distance (ft) } & 0 & 16 & 64 & 144 & 256 & 400 \\ \hline \end{array} $$ (a) Make a scatterplot of the ordered pairs determined by (time, velocity) and (time, distance) in the same vicwing rectangle \([-1,6,1]\) by \([-10,450,50]\) (b) Find a function \(v\) that models the velocity. (c) The distance is modeled by \(d(x)=a x^{2} .\) Find \(a\) (d) Find the time when the distance is 200 feet. Find the velocity at this time.

Use a table to solve. \(\frac{1}{8} x^{2}+x+2 \geq 0\)
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