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Chapter 2: Problem 3

Identify the underlying basic function, and use transformations of the basic function to sketch the graph of the given function. $$f(x)=\sqrt{x}-2$$

Short Answer

Expert verified
The graph of the function \( f(x)=\sqrt{x}-2 \) is a square root curve that has been shifted down 2 units and starts from the point (0, -2).

Step by step solution

01

Identify the Basic Function

The basic function here is the square root function, \( \sqrt{x} \). The graph of this function is a curve that starts from the origin (0,0) and increases as x increases. It is only defined for x鈮�0.
02

Identify the Transformation

The transformation is represented by the -2 in the function \( \sqrt{x}-2 \). This is a vertical shift down by 2 units, which means every point on the graph of \( \sqrt{x} \) will be moved 2 units down to form the graph of \( \sqrt{x}-2 \).
03

Sketch the graph

Start with the graph of the basic function \( \sqrt{x} \) and move each point on the graph down by 2 units. The resulting graph still starts at x=0, but at y=-2 due to the vertical shift, and continues to increase as x increases.

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Key Concepts

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

Understanding the Square Root Function
The square root function is the fundamental function in the equation \( f(x) = \sqrt{x} \). It forms a curve that starts at the origin, which is the point (0,0), and extends upwards to the right. This particular function is only defined for values where \( x \geq 0 \). The graph demonstrates a gentle curve that tends to rise more slowly as \( x \) becomes larger.

Key features of the square root function include:
  • A starting point, or vertex, at (0,0).
  • The curve extends to the right along the x-axis.
  • As \( x \) increases, \( \sqrt{x} \) increases as well, but at a decreasing rate.
Understanding these properties helps when we need to apply transformations, as it gives us a baseline to work from.
Exploring Vertical Shifts
Vertical shifts are transformations that move the graph of a function up or down along the y-axis. In the function \( f(x) = \sqrt{x} - 2 \), the \(-2\) represents a vertical shift. This specific shift moves the entire graph of the basic function \( \sqrt{x} \) downward by 2 units.

When applying a vertical shift:
  • Every y-coordinate on the graph of the original function is adjusted by the shift value.
  • If the shift value is negative, the graph moves downward. If positive, the graph moves upward.
  • Importantly, the shape of the graph does not change 鈥� only its position does.
In our example, starting from the point (0,0) on the square root function, after the vertical shift of -2, the new starting point becomes (0,-2). This means the entire graph just moves down by 2 units along the y-axis.
Steps for Graph Sketching
Graph sketching involves drawing the basic shape of a function and applying any necessary transformations. Here's how to sketch the graph of \( f(x) = \sqrt{x} - 2 \):

  • Begin with the graph of the square root function \( \sqrt{x} \), which starts at (0,0).
  • Recognize the transformation. Here, it is a downward vertical shift by 2 units.
  • Move each point on the graph of \( \sqrt{x} \) downward by 2 units. For example, the point (0,0) will become (0,-2), and the point (1,1) will become (1,-1).
  • Sketch the resulting graph. It should maintain the same shape as \( \sqrt{x} \), just moved down by 2 units.
This process makes it easier to visualize how the graph of the transformed function appears and provides a straightforward method for sketching it.

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Most popular questions from this chapter

In Exercises \(87-96,\) find two functions \(f\) and \(g\) such that \(h(x)=(f \circ g)(x)=f(g(x)) .\) Answers may vary. $$h(x)=\sqrt[3]{4 x^{2}-1}$$

The range of a quadratic function \(g(x)=a x^{2}+b x+c\) is given by \((-\infty, 2] .\) Is \(a\) positive or negative? Justify your answer.

A ball is thrown directly upward from ground level at time \(t=0\) ( \(t\) is in seconds). At \(t=3,\) the ball reaches its maximum distance from the ground, which is 144 feet. Assume that the distance of the ball from the ground (in feet) at time \(t\) is given by a quadratic function \(d(t) .\) Find an expression for \(d(t)\) in the form \(d(t)=a(t-h)^{2}+k\) by performing the following steps. (a) From the given information, find the values of \(h\) and \(k\) and substitute them into the expression \(d(t)=a(t-h)^{2}+k\) (b) Now find \(a\). To do this, use the fact that at time \(t=0\) the ball is at ground level. This will give you an equation having just \(a\) as a variable. Solve for \(a\) (c) Now, substitute the value you found for \(a\) into the expression you found in part (a). (d) Check your answer. Is (3,144) the vertex of the associated parabola? Does the parabola pass through (0,0)\(?\)

This set of exercises will draw on the ideas presented in this section and your general math background. Explain what is wrong with the following steps for solving a radical equation. $$\begin{aligned}\sqrt{x+1}-2 &=0 \\\\(x+1)+4 &=0 \\\x &=-5\end{aligned}$$

Sketch a graph of the quadratic function, indicating the vertex, the axis of symmetry, and any \(x\)-intercepts. $$f(t)=-t^{2}-1$$
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