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Chapter 2: Problem 45

$$\text {Use transformations of graphs to sketch a graph of } y=f(x) \text { by }$$ $$f(x)=-\sqrt{1-x}$$

Short Answer

Expert verified
The graph of \( y = -\sqrt{1-x} \) is a downward curve starting at (1, 0), going to the left.

Step by step solution

01

Identify the Base Function

The base function here is \( y = \sqrt{x} \). This is the square root function, which is typically defined for \( x \geq 0 \). Its graph is a curve that starts from the origin (0,0) and rises slowly to the right.
02

Apply Horizontal Shift

In the given function \( f(x) = -\sqrt{1-x} \), the term \( (1-x) \) indicates a horizontal shift. Rewriting it as \( -(x-1) \) shows that there is a shift to the right by 1 unit.
03

Apply Reflection and Vertical Shift

The negative sign outside the square root in \( f(x) = -\sqrt{1-x} \) indicates a reflection over the x-axis, which flips the base function upside down.
04

Determine the Domain

The expression inside the square root, \( 1-x \), must be non-negative. Thus, solve \( 1-x \geq 0 \) to find that the domain is \( x \leq 1 \).
05

Sketch the Graph

Using the transformations applied: start with the base curve \( \sqrt{x} \), shift it to the right by 1 unit, reflect it over the x-axis, and only consider the portion where \( x \leq 1 \). Begin at point (1,0) and curve downwards to the left.

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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, denoted by \(y = \sqrt{x}\), is a well-known fundamental mathematical function.
It generates a curve that begins at the origin point (0,0) and progresses upward in a gentle curve to the right as the value of \(x\) increases. Since it involves taking the square root, the function is only defined for values where \(x\) is greater than or equal to zero.
In graphing this function, you'll notice that the push upward becomes less steep, creating a gentle, continuous curve.
Horizontal Shift
A horizontal shift is a type of transformation that moves a graph left or right from its position. In mathematical terms, when you see an expression like \((x - c)\) inside a function, it means the graph is shifted \(c\) units. If \(c\) is positive, shift right; if \(c\) is negative, shift left.
Applying this to our function \(f(x) = -\sqrt{1-x}\), which can be rewritten as \(-\sqrt{-(x-1)}\), we interpret the \(-x + 1\) term as a horizontal shift. Rewrite it as \(-(x-1)\) to identify this graph shifts to the right by 1 unit.
Reflection
A reflection transforms a graph to mirror its shape across an axis. For functions, a reflection over the x-axis flips the entire graph vertically.
To reflect a function across the x-axis, multiply the entire function by -1. In the case of \(f(x) = -\sqrt{1-x}\), the negative sign outside the square root indicates such a reflection.
This means our original curve, which typically rises, will now fall as it flips upside down.
Domain of Function
The domain of a function comprises all possible input values (x-values) for which the function is defined. For a square root function, the expression inside the square root must be zero or positive.
For \(f(x) = -\sqrt{1-x}\), solve \(1-x \geq 0\) to find that \(x\) must be less than or equal to 1.
This condition ensures that the square root takes real number values, maintaining the function's validity. Hence, the domain is \(x \leq 1\).

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Most popular questions from this chapter

The table below shows the acreage, in millions, of the total of com and soybeans harvested annually in the United States. In the table, \(x\) represents the year and \(f\) computes the total number of acres for these two crops. The function \(g\) computes the number of acres for corn only. $$\begin{array}{c|c|c|c|c}\hline x & 2013 & 2014 & 2015 & 2016 \\\\\hline f(x) & 175.1 & 176.4 & 174.0 & 177.8 \\\\\hline g(x) & 97.4 & 91.6 & 88.9 & 94.1\end{array}$$ (a) Make a table for a function \(h\) that is defined by the equation \(h(x)=f(x)-g(x)\) \(\Rightarrow\) (b) Interpret what \(h\) computes.

Each function is either even or odd Evaluate \(f(-x)\) to determine which situation applies. $$f(x)=\sqrt{x^{2}+1}$$

Solve the following. Complete the following. (a) Write a function \(F\) that converts \(x\) miles to feet. (b) Write a function \(I\) that converts \(x\) feet to inches. (c) Write a formula for the composition \((I \circ F)(x)\) (d) Explain what \((I \circ F)(x)\) calculates.

Use the analyric method of Example 3 to determine whether the graph of the given function is symmetric with respect to the \(y\) -axis, symmetric with respect to the origin, or neither. Use \(a\) calculator and the standand window to support your conclusion. $$f(x)=x^{3}-3 x$$

Determine the difference quotient \(\frac{f(x+h)-f(x)}{h}\) (where \(h \neq 0\) j for each function \(f\). Simplify completely. $$f(x)=x^{2}+2 x$$
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