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Chapter 2: Problem 21

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.) $$f(x)=c \sqrt{4-x^{2}}, \quad c=-2,1,3$$

Short Answer

Expert verified
Three semicircles with c affect stretch, compression, and reflection on the coordinate plane.

Step by step solution

01

Identify the base function

The base function is given by \( y = \sqrt{4-x^2} \). This represents a semicircle with a radius of 2 centered at the origin on the x-axis.
02

Understand transformations by c-values

The value of \( c \) will stretch or compress the graph vertically and can also reflect it across the x-axis if \( c \) is negative. For \( c = -2, 1, 3 \), this represents reflection and stretching/compressing for each \( c \).
03

Graph for c = 1

The function \( f(x) = \sqrt{4-x^2} \) is the semicircle from \( -2 \) to \( 2 \). It's already given for \( c = 1 \), so we plot the half-circle spanning from \(( -2, 0 )\) through \((0, 2)\) to \(( 2, 0 )\).
04

Graph for c = -2

For \( c = -2 \), we reflect the semicircle across the x-axis and stretch it vertically by a factor of 2. The new function is \( f(x) = -2 \sqrt{4-x^2} \). This new graph spans from \(( -2, 0 )\) down to \((0, -4)\) and back up to \(( 2, 0 )\).
05

Graph for c = 3

For \( c = 3 \), the semicircle is stretched vertically by a factor of 3 but is not reflected. The new function is \( f(x) = 3 \sqrt{4-x^2} \) resulting in the top semicircle from \(( -2, 0 )\) up through \(( 0, 6 )\) to \(( 2, 0 )\).
06

Combine the graphs

All these transformations should be drawn on the same coordinate plane to showcase the transformations due to different \( c \) values. This results in three semicircles corresponding to \( c = -2, 1, 3 \).

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Key Concepts

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

Semicircle Function
A semicircle function is a familiar quadratic function that forms a half-circle on the coordinate plane. In the case of the exercise, you're dealing with the function:
  • \( y = \sqrt{4-x^2} \) represents the top half of a circle with a radius of 2.
    • This is derived from the equation of a circle, \( x^2 + y^2 = 4 \), centered at the origin.
    • The positive square root gives the upper semicircle.
The semicircle spans from \( x = -2 \) to \( x = 2 \) on the horizontal axis, with the highest point at \( (0, 2) \). It's a useful function for illustrating basic characteristics like symmetry and transformations. Therefore, it often appears in exercises involving graphical transformations.
Vertical Stretching
Vertical stretching is an important transformation that influences the height of a graph without changing it horizontally. This transformation is controlled by the value of \( c \) in the function \( f(x) = c \sqrt{4-x^2} \). Let's look at how this plays out:
  • For \( c = 1 \), the graph maintains its original size, representing the basic semicircle.
  • When \( c = 3 \), the graph stretches vertically by a factor of 3, making it three times taller, with the new highest point at \( (0, 6) \).
The concept becomes visual by comparing these changes on a graph. The semicircle grows taller while keeping the same base width, illustrating how vertical stretching alters the overall appearance of the function.
Reflection on X-Axis
A reflection on the x-axis flips a graph upside down. In mathematical terms, it is represented by a negative multiplier. Here, when \( c = -2 \), this reflection takes place:
  • The original semicircle, \( y = \sqrt{4-x^2} \), turns into \( y = -2 \sqrt{4-x^2} \).
  • This transformation flips the semicircle from above the x-axis to below it.
  • Additionally, the semicircle is vertically stretched by a factor of 2.
  • The lowest point reaches \( (0, -4) \).
Reflection is a key transformation that changes the orientation of the graph, creating an inverse visual. By reflecting over the x-axis, you've altered both its position and the graph's perceived direction of growth.
Symmetry in Graphs
Symmetry in graphs is an essential concept that helps in understanding the balanced appearance of functions around a line or point. In our exercise, symmetry simplifies plotting:
  • The base semicircle function, \( y = \sqrt{4-x^2} \), is symmetric about the y-axis.
  • This means if you were to fold the graph along the y-axis, both halves would overlap perfectly.
The property of symmetry makes it easier to predict how functions will change under various transformations, such as stretching and reflection. By focusing on one side of the graph and knowing it will mirror on the other, drawing these transformations becomes simpler, improving the visualization of the entire graph.

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