Table of contents

0. Review of College Algebra4h 43m
1. Measuring Angles40m
2. Trigonometric Functions on Right Triangles2h 5m
3. Unit Circle1h 19m
4. Graphing Trigonometric Functions1h 19m
5. Inverse Trigonometric Functions and Basic Trigonometric Equations1h 41m
6. Trigonometric Identities and More Equations2h 34m
7. Non-Right Triangles1h 38m
8. Vectors2h 25m
9. Polar Equations2h 5m
10. Parametric Equations1h 6m
11. Graphing Complex Numbers1h 7m

4. Graphing Trigonometric Functions

Graphs of the Sine and Cosine Functions

Trigonometry

4. Graphing Trigonometric Functions

Graphs of the Sine and Cosine Functions

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2:14 minutes

Problem 75

Textbook Question

Graph each function. See Examples 6–8.ƒ(x) = √-x

Verified step by step guidance

Step 1: Identify the function type. The function \( f(x) = \sqrt{-x} \) is a square root function, but it involves a negative sign inside the square root.

Step 2: Determine the domain of the function. Since the square root function is only defined for non-negative numbers, \( -x \geq 0 \) implies \( x \leq 0 \). Therefore, the domain of \( f(x) \) is all real numbers less than or equal to zero.

Step 3: Analyze the range of the function. The square root function outputs non-negative values, so \( f(x) \geq 0 \).

Step 4: Consider the transformation. The function \( f(x) = \sqrt{-x} \) can be seen as a reflection of the basic square root function \( \sqrt{x} \) across the y-axis.

Step 5: Sketch the graph. Start by plotting key points such as \( (0, 0) \) and \( (-1, 1) \), and draw a smooth curve that reflects the shape of the square root function, extending to the left along the x-axis.

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Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

Domain of a Function

The domain of a function refers to the set of all possible input values (x-values) for which the function is defined. For the function ƒ(x) = √-x, the expression under the square root must be non-negative, meaning -x ≥ 0 or x ≤ 0. This indicates that the function is only defined for x-values less than or equal to zero.

Graphing Square Root Functions

Graphing square root functions involves plotting points based on the function's output for given inputs. For ƒ(x) = √-x, the graph will consist of points where x is non-positive, and the output will be real numbers. The graph will start at the origin (0,0) and extend leftward, forming a curve that rises as x approaches zero.

Transformations of Functions

Transformations of functions involve shifting, reflecting, stretching, or compressing the graph of a function. In the case of ƒ(x) = √-x, the negative sign indicates a reflection across the y-axis compared to the standard square root function ƒ(x) = √x. Understanding these transformations helps in accurately sketching the graph based on the original function.