Problem 21 - Algebra and Trigonometry with Analytic Geometry

Question

Exer. 13-26: 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.) $$ f(x)=c \sqrt{4-x^{2}} ; \quad c=-2,1,3 $$

Step-by-Step Solution

Answer

Graph three semicircles with varying scalings and reflections for c values.

Step 1: Understand the base function

The base function, without the constant multiplier, is defined as \( f(x) = \sqrt{4 - x^2} \). This is the equation of a semicircle with radius 2 centered at the origin on the x-axis.

Step 2: Effect of different values of c

The constant \( c \) controls the vertical scaling and direction of the semicircle. \( c = 1 \) keeps the semicircle upright, \( c = -2 \) reflects it down and doubles its size, and \( c = 3 \) stretches it upward by a factor of three.

Step 3: Graphing for c = 1

For \( c = 1 \), graph \( g(x) = \sqrt{4 - x^2} \). This remains an upright semicircle with radius 2.

Step 4: Graphing for c = -2

For \( c = -2 \), graph \( h(x) = -2\sqrt{4 - x^2} \). The semicircle is reflected over the x-axis and its height is doubled.

Step 5: Graphing for c = 3

For \( c = 3 \), graph \( j(x) = 3\sqrt{4 - x^2} \). The semicircle is stretched upward, tripling its height compared to the base graph.

Key Concepts

Semicircle Graph

The concept of a semicircle graph comes into play when you have a function like \( f(x) = \sqrt{4 - x^2} \). This function is derived from the equation of a circle, but it only represents the upper half. So, it's called a semicircle. For a complete circle, you'd have \( x^2 + y^2 = r^2 \), which, in this case, is \( 4 \), making the radius 2.
- The semicircle graph is centered at the origin, \( (0,0) \).- The radius is 2, meaning it reaches from \(-2\) to \(2\) on the \(x\)-axis.- It sits above the \(x\)-axis, creating a curved, smooth arc.
Semicircle graphs are important in visualizing constraints or portions of larger circles. Understanding this gives you a foundation for discussing other transformations like scaling and reflecting.

Vertical Scaling

Vertical scaling is all about altering the vertical size of the graph. You handle this through the multiplier, \( c \), in the function \( f(x) = c \sqrt{4 - x^2} \).
- When \( c = 1 \), the semicircle's height remains unchanged.- If \( c = 3 \), the semicircle size vertically triples as you multiply the heights by 3.- For \( c = -2 \), while still vertically scaling, the negative sign induces reflection.
Think of vertical scaling like adjusting the volume on a speaker; you're just changing the level at which the graph is "heard" along the \(y\)-axis. This concept is critical when altering how functions behave visually on a graph.

Function Reflection

Function reflection involves flipping the graph over a reference line, typically the \(x\) or \(y\)-axis. With our exercise, function reflection comes into play when \( c \) is negative.
- For example, \( c = -2 \) results in the function \( h(x) = -2\sqrt{4 - x^2} \).- This reflects the graph over the \(x\)-axis, flipping it downward.- Consequently, everything that was above the \(x\)-axis moves below, maintaining the semicircle shape.
Reflections are like gazing into a mirror. They give us an inverted image. In graph terms, it's inverting the output values, crucial for grasping how negative values affect graph geometry.

Graph Shifting

Graphs can also undergo shifting, where they move horizontally or vertically across the coordinate plane. Although our original exercise doesn't require shifting, understanding it complements other transformations.
- Horizontal shifting occurs when parts of the function, say \( (x-2) \), change the graph's position left or right.- Vertical shifts add or subtract constants directly from the function, like \( f(x) + 3 \), lifting it up or down.- Neither of these shifts expand or compress the graph, just position it differently.
When working with more complex functions or combinations of transformations, knowing how to shift graphs comes in handy. It provides control over function placement within the plane and can simplify comparing multiple graphs.