Problem 16 Sketch, on the same coordinate p... [FREE SOLUTION]

Chapter 1: Problem 16

Sketch, on the same coordinate plane, the graphs of $f$ for the given values of $c$. (Make use of symmetry, vertical shifts, horizontal shifts, stretching, or reflecting.) $$ f(x)=\sqrt{9-x^{2}}+c ; \quad c=0,1,-3 $$

Short Answer

Expert verified

Sketch semicircles for $c=0$, $c=1$, and $c=-3$, each vertically shifted by $c$.

Step by step solution

Identifying the Base Function

The base function is given by $ f(x) = \sqrt{9 - x^2} $. This function represents the upper half of a circle with radius 3 centered at the origin. The domain is $-3 \leq x \leq 3$ as the square root of a negative number isn't defined in the real numbers.

Graphing the Base Function

Sketch the semicircle defined by $ f(x) = \sqrt{9 - x^2} $ which has endpoints at $(-3, 0)$ and $(3, 0)$ and a maximum point at $(0, 3)$. This represents the graph of the function when $ c = 0 $.

Exploring the Effect of $ c = 1 $

Add $1$ to the function: $ f(x) = \sqrt{9 - x^2} + 1 $. This results in shifting the semicircle graph up by 1 unit. The new maximum point is at $(0, 4)$ and the endpoints are at $(-3, 1)$ and $(3, 1)$.

Exploring the Effect of $ c = -3 $

Subtract $3$ from the function: $ f(x) = \sqrt{9 - x^2} - 3 $. This results in shifting the semicircle graph down by 3 units. The new maximum point is at $(0, 0)$ and the endpoints are at $(-3, -3)$ and $(3, -3)$.

Combining the Graphs

Draw all three graphs on the same coordinate plane: the original semicircle for $c = 0$, the shifted graph up for $c = 1$, and the shifted graph down for $c = -3$. Label each graph accordingly.

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Key Concepts

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

Vertical Shifts

In the graphing world, a vertical shift refers to moving a graph up or down along the y-axis. It's like taking a curtain and pulling it up or letting it down a bit. This shifting doesn't change the shape of the graph, just where it sits vertically.

When we add a constant to a function, for example, adding or subtracting a certain value from a function, it results in this vertical shift. In our given function, which is the semicircle function, the transformations occur through different values of $ c $.

For $ c = 1 $, we shift the entire graph up by 1 unit, making sure the entire pattern stays intact.
For $ c = -3 $, we shift the graph down by 3 units. Instead of being at the top of the coordinate plane, it pushes the graph downward.

These vertical shifts help us analyze how changes to a function impact its graph's position without changing its original formation.

Semicircle Graph

A semicircle graph is simply half of a circle. In this specific exercise, we are looking at the upper half. Imagine cutting a whole circle across its diameter and discarding the bottom half.

The equation $ f(x) = \sqrt{9 - x^2} + c $ describes this semicircle aspect perfectly. The $ \sqrt{9 - x^2} $ part confines our graph to the upper half due to the square root function only yielding non-negative outputs.

The endpoints of the semicircle graph are at $(-3, 0)$ and $(3, 0)$.
The topmost point (or the peak) is at the middle of the endpoints, at $(0, 3)$ when $ c = 0 $.

Understanding semicircle graphs is important, especially when considering phenomena that symmetrically spread around a central point or have circular characteristics.

Radius of Circle

The radius of a circle is the distance from the center of the circle to any point on its perimeter. It is also half the diameter of the circle.

For our function $ f(x) = \sqrt{9 - x^2} $, the expression inside the square root indicates a circle. Specifically, the equation is in the form of a circle with center at the origin, $(0, 0)$, given by $ x^2 + y^2 = r^2 $. Here we have $ x^2 + y^2 = 9 $.

The term $ 9 $ represents $ r^2 $, so the radius $ r $ is $ \sqrt{9} = 3 $.
This radius dictates the size of our semicircle, spanning horizontally from $-3$ to $3$.

Grasping this concept of the radius is crucial in understanding how it regulates the spread of our semicircle graph.

Real Number Domain

The real number domain for a function is the set of all possible input values (x-values) that do not cause any mathematical rule-breaking. In this scenario, it means numbers can be inputted without leading to an undefined or non-real result.

For the equation $ f(x) = \sqrt{9 - x^2} $, we're limited in our domain because of the square root function.

The values inside our square root, $9 - x^2$, must always remain non-negative, since the square root of negative numbers is not real.
This condition establishes that $-3 \leq x \leq 3 $. The function is defined for values within this range.

Real number domains ensure that our graph is accurately drawn and comprehensible within feasible limitations.

91影视

Sketch, on the same coordinate plane, the graphs of \(f\) for the given values of \(c\). (Make use of symmetry, vertical shifts, horizontal shifts, stretching, or reflecting.) $$ f(x)=\sqrt{9-x^{2}}+c ; \quad c=0,1,-3 $$

Short Answer

Step by step solution

Identifying the Base Function

Graphing the Base Function

Exploring the Effect of \( c = 1 \)

Exploring the Effect of \( c = -3 \)

Combining the Graphs

Key Concepts

Vertical Shifts

Semicircle Graph

Radius of Circle

Real Number Domain

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