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Chapter 2: Problem 48

Use transformations of graphs to sketch a graph of \(y=f(x)\) by hand. Do not use a calculator. $$f(x)=2+\sqrt{-(x-3)}$$

Short Answer

Expert verified
The graph of \( f(x) = 2 + \sqrt{-(x-3)} \) is a square root curve reflected across the y-axis, shifted 3 units right, and 2 units up, starting at (3,2).

Step by step solution

01

Identify the Base Function

The base function is the square root function, typically written as \( y = \sqrt{x} \). This graph has its vertex at the origin (0,0) and moves to the right in a positive curve.
02

Consider the Inside Transformation

The function is given as \( y = \sqrt{-(x-3)} \). The term \(-(x-3)\) indicates a horizontal transformation. The negative sign means a reflection about the y-axis, and the \(x-3\) indicates a horizontal shift to the right by 3 units.
03

Apply Horizontal Reflection and Shift

Reflect the base function \( y=\sqrt{x} \) across the y-axis. This creates \( y=\sqrt{-x} \), which is defined for \( x \leq 0 \) only. Next, shift this graph horizontally 3 units to the right, changing the vertex from \((0,0)\) to \((3,0)\). This gives the graph of \( y = \sqrt{-(x-3)} \) which is defined for \( x \leq 3 \).
04

Consider the Vertical Shift

Finally, the function \( y = \sqrt{-(x-3)} + 2 \) indicates a vertical shift upwards by 2 units. Adjust every point on the graph up by 2 to account for this.
05

Sketch the Transformed Graph

Start with the vertex at (3,2). The graph extends horizontally towards \( x = -\infty \) while just touching the y-axis at the point where it was shifted vertically, making it a transformed square root curve that is reflected horizontally.

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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 mathematical function represented by \( y = \sqrt{x} \). Its primary characteristic is that it only outputs non-negative values because the square root of any non-negative number is also non-negative. This function forms the base for many graph transformations.

Visually, the square root function begins at the origin \( (0,0) \) and extends to the right, forming a gentle upward curve. This is because as \( x \) increases, \( \sqrt{x} \) also increases, but at a decreasing rate.
  • The graph only exists for \( x \geq 0 \).
  • The shape is a half parabola opening upwards.
Understanding this base form helps in visualizing how the graph changes with different transformations.
Horizontal Reflection
A horizontal reflection flips the graph of a function over the y-axis. In mathematical terms, for a function \( y = f(x) \), this transformation would be represented as \( y = f(-x) \).

In the case of the square root function, this reflection results in the curve moving leftward, maintaining its half-parabolic shape.
  • For \( y = \sqrt{-x} \), the domain changes to \( x \leq 0 \), because the square root is only defined for non-negative values.
  • This reflection causes the curve to start at the origin and extend left.
Thus, a horizontal reflection affects the direction of the curve and the domain of the function.
Vertical Shift
A vertical shift involves moving a graph up or down along the y-axis. This transformation can be expressed as \( y = f(x) + c \), where \( c \) is the number of units the graph shifts. If \( c > 0 \), the graph moves upward; if \( c < 0 \), it moves downward.

In our function, the addition of 2 indicates a vertical shift upwards by 2 units.
  • Every point moves up by 2 units.
  • The entire graph is raised without changing its shape.
This shift does not alter the domain or range besides increasing each y-value by 2.
Horizontal Shift
Horizontal shifts involve moving the graph either left or right along the x-axis. This is achieved by adding or subtracting a constant from the variable \( x \) in the function, represented as \( y = f(x-h) \). Here, \( h \) represents the horizontal shift.

In the given function \( y = \sqrt{-(x-3)} \), the \( x-3 \) indicates a shift 3 units to the right:
  • The graph moves right, positioning the vertex from \( (0,0) \) to \( (3,0) \).
  • Initially defined for \( x \leq 0 \), the domain adjusts to \( x \leq 3 \).
Horizontal shifting helps reposition the graph while maintaining the same orientation.

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Most popular questions from this chapter

Each inequality describes the range of average monthly temperatures \(T\) in degrees Fahrenheit at a certain location. (a) Solve the inequality. (b) Interpret the result. \(|T-61.5| \leq 12.5,\) Buenos Aires, Argentina

For certain pairs of functions \(f\) and \(g,(f \circ g)(x)=x\) and \((g \circ f)(x)=x .\) Show that this is true for the pairs. $$f(x)=4 x+2, g(x)=\frac{1}{4}(x-2)$$

Consider the function h as defined. Find functions \(f\) and \(g\) such that \((f \circ g)(x)=h(x) .\) (There are several possible ways to do this.) $$h(x)=(2 x-3)^{3}$$

An equation of the form \(|f(x)|=|g(x)|\) is given. (a) Solve the equation analytically and support the solution graphically. (b) Solve \(|f(x)|>|g(x)|\) (c) Solve \(|f(x)|<|g(x)|\) $$|5 x-6|=|-5 x+6|$$

Consider the function h as defined. Find functions \(f\) and \(g\) such that \((f \circ g)(x)=h(x) .\) (There are several possible ways to do this.) $$h(x)=\sqrt{x^{2}-1}$$
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