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Chapter 2: Problem 18

Sketch, on the same coordinate plane, the graphs of \(f\) for the given values of \(c\). (Make use of symmetry, shifting, stretching, compressing, or reflecting.) $$f(x)=\sqrt{9-x^{2}}+c ; \quad c=-3,0,2$$

Short Answer

Expert verified
Graph semicircles centered at y = 0, -3, and 2 with endpoints at x = -3 and x = 3.

Step by step solution

01

Understand the Function

The function given is \( f(x) = \sqrt{9 - x^2} + c \). It represents a semicircle with radius 3 centered at the origin on the x-axis, because \( \sqrt{9 - x^2} \) is the equation of a semicircle. The parameter \( c \) vertically shifts this semicircle.
02

Graph the Parent Function

Graph \( f_0(x) = \sqrt{9 - x^2} \) first, which represents the top half of a circle with radius 3 centered at the origin, having domain \(-3 \le x \le 3\) and range \(0 \le y \le 3\).
03

Apply Vertical Shift for \(c = 0\)

For \( c = 0 \), the function is \( f(x) = \sqrt{9 - x^2} \). This is the same as the parent function and is already graphed in the previous step.
04

Apply Vertical Shift for \(c = -3\)

For \( c = -3 \), the function becomes \( f(x) = \sqrt{9 - x^2} - 3 \). This shifts the parent semicircle down by 3 units, so now the range is \(-3 \le y \le 0\).
05

Apply Vertical Shift for \(c = 2\)

For \( c = 2 \), the function becomes \( f(x) = \sqrt{9 - x^2} + 2 \). This shifts the parent semicircle up by 2 units, resulting in a new range of \(2 \le y \le 5\).
06

Sketch the Combined Graphs

Plot all three functions on the same coordinate plane. They are semicircles centered at the x-axis with respective vertical shifts determined by \( c = 0, -3, \) and \(2\). Their endpoints remain \( x = -3 \) and \( x = 3 \) for each function.

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Key Concepts

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

Vertical Shifts
Vertical shifts are a type of transformation that involve moving the graph of a function up or down along the y-axis. This is done by adding or subtracting a constant value, noted as \( c \), to the function. In the case of the function \( f(x) = \sqrt{9 - x^2} + c \), \( c \) determines how much the graph of the semicircle is shifted vertically.
For example:
  • When \( c = 0 \), there is no vertical shift, so the semicircle remains in its original position.
  • For \( c = -3 \), the semicircle shifts down 3 units, altering the range of the function.
  • If \( c = 2 \), the shift is 2 units upwards.
Vertical shifts do not affect the shape or horizontal position of the graph, only its elevation on the y-axis. It's essential to adjust the range of the function accordingly to describe precisely where the graph will be located after the shift is applied.
Semicircle Equations
The semicircle equation \( y = \sqrt{a^2 - x^2} \) represents the top half of a circle with radius \( a \). In this exercise, the equation is \( y = \sqrt{9 - x^2} \), which forms a semicircle with radius 3. This semicircle is centered on the x-axis at the origin, stretching from \( x = -3 \) to \( x = 3 \).
Here are some key aspects of semicircle equations:
  • The domain is restricted by the radius, ensuring \(-a \leq x \leq a\).
  • The range starts at 0 and goes up to the radius since it's only half of a full circle.
  • This equation inherently has symmetry: it mirrors across the y-axis.
Recognizing the semicircle form is vital for understanding its behavior when transformed by vertical shifts or other manipulations.
Function Sketching
Sketching functions involves a few key steps to capture essential features such as shape, position, and transformations. For our function \( f(x) = \sqrt{9 - x^2} + c \), consider these aspects:
  • Start by sketching the parent function \( y = \sqrt{9-x^2} \), a semicircle centered at the origin.
  • Use symmetry to assist your sketching; the graph reflects over the y-axis.
  • Apply vertical shifts based on the value of \( c \) to sense the function elevation.
  • Mark endpoints on the x-axis at \( x = -3 \) and \( x = 3 \).
By plotting these elements on the same coordinate plane, you'll effectively visualize how changes in \( c \) affect the graph's position, allowing clearer comparisons and a better understanding of transformation impacts.
Domains and Ranges
Domains and ranges are fundamental concepts that define where a function exists and the values it can take. Understanding these ideas is crucial for sketching correct graphs.
  • The domain of our semicircle function \( f(x) = \sqrt{9 - x^2} + c \) is limited by the circle equation and is \(-3 \le x \le 3\). This shows that the function only exists within this interval on the x-axis.
  • The range depends on the vertical shift value \( c \).
  • For \( c = 0 \), the range is \(0 \le y \le 3\).
  • With \( c = -3 \), it adjusts to \(-3 \le y \le 0\).
  • For \( c = 2 \), it becomes \(2 \le y \le 5\).
Identifying these constraints ensures accuracy in graph drawing and comprehension of how the function behaves over different values.

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Most popular questions from this chapter

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