Question

In Exercises 55–58, graph each function, not by plotting points, but by starting with the graph of one of the standard functions presented in Figures 1.15–1.17, and applying an appropriate transformation.
y = - √(1 + x/2)

Accepted Answer

Identify the base function: The base function here is $$ y = \sqrt{x} . $$This is a standard square root function. Apply the transformation inside the square root: The expression inside the square root is $$ 1 + \frac{x}{2} . $$This represents a horizontal shift. Specifically, $$ x  is $$replaced by $$ x + 2 $$, which shifts the graph to the left by 2 units. Apply the vertical reflection: The negative sign in front of the square root, $$ y = -\sqrt{1 + \frac{x}{2}} $$, indicates a reflection across the x-axis. This means that all y-values of the transformed function will be the opposite of the y-values of the base function. Consider the domain: The domain of the base function $$ y = \sqrt{x}  is$$ $$ x \geq 0 . $$For the transformed function $$ y = -\sqrt{1 + \frac{x}{2}} $$, the expression inside the square root must be non-negative, so $$ 1 + \frac{x}{2} \geq 0 . $$Solve this inequality to find the domain of the transformed function. Graph the function: Start with the graph of $$ y = \sqrt{x} $$, apply the horizontal shift to the left by 2 units, and then reflect the graph across the x-axis. Ensure the graph only includes x-values within the domain found in the previous step.

In Exercises 55–58, graph each function, not by plotting points, but by starting with the graph of one of the standard functions presented in Figures 1.15–1.17, and applying an appropriate transformation.
y = - √(1 + x/2)

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Concetti chiave

Standard Functions

Function Transformations

Square Root Function

In Exercises 55–58, graph each function, not by plotting points, but by starting with the graph of one of the standard functions presented in Figures 1.15–1.17, and applying an appropriate transformation.y = - √(1 + x/2)

Risposta video verificata per un problema simile:

Concetti chiave

Standard Functions

Function Transformations

Square Root Function

In Exercises 55–58, graph each function, not by plotting points, but by starting with the graph of one of the standard functions presented in Figures 1.15–1.17, and applying an appropriate transformation.
y = - √(1 + x/2)