Problem 79

Question

Begin by graphing the square root function, $f(x)=\sqrt{x} .$ Then use transformations of this graph to graph the given function. $$g(x)=2 \sqrt{x+2}-2$$

Step-by-Step Solution

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Answer

To graph $g(x)=2 \sqrt{x+2}-2$, start with the graph of $f(x)=\sqrt{x}$ and then shift it 2 units to the left, stretch it vertically by a factor of 2 and finally shift it down by 2 units.

1Step 1: Graph the base function

Start by graphing the original function $f(x) = \sqrt{x}$. Note that this is a basic square root function beginning at the origin (0,0) and slowly increasing as x gets larger.

2Step 2: Apply horizontal shift

The $+2$ within the square root function represents a horizontal shift. The graph of $f(x)=\sqrt{x+2}$ is a shift of the graph $f(x)=\sqrt{x}$ 2 units to the left.

3Step 3: Apply vertical shift and scaling

The $2$ outside the square root function and the $-2$ at the end indicate a scaling and a vertical shift respectively. The 2 multiplied with the square root function stretches the graph vertically by a factor of 2 and the $-2$ at the end shifts the graph down by 2 units. This will give you the graph of $g(x)=2\sqrt{x+2}-2$.

Key Concepts

Graph TransformationsHorizontal ShiftVertical ShiftFunction Graphing

Graph Transformations

Graph transformations are changes made to the position, shape, or size of a graph. They help us understand how different aspects of a function affect its graph. When dealing with the graph of a square root function like $ f(x) = \sqrt{x} $, transformations make it possible to predict how the graph will change under different conditions.
To transform a graph, you can apply various modifications like shifting it up, down, left, or right, stretching or compressing it, or even reflecting it.

Shifting: Moving the graph horizontally or vertically.
Scaling: Increasing or decreasing the size of the graph.
Reflection: Flipping the graph over a coordinate axis.

Understanding these transformations helps predict the shape and position of more complex functions from simpler base graphs.

Horizontal Shift

A horizontal shift moves the entire graph of a function left or right. This is typically the result of adding or subtracting a constant inside the function argument. For square roots, consider the function $ f(x) = \sqrt{x} $.
When you transform it into $ f(x) = \sqrt{x+2} $, the graph of $ \sqrt{x} $ shifts 2 units to the left.

Left Shift: Adding inside the function $(\sqrt{x+c})$ moves the graph to the left by $ c $ units.
Right Shift: Subtracting inside the function $(\sqrt{x-c})$ shifts the graph to the right by $ c $ units.

This means that every point on the graph moves in the horizontal direction, while maintaining the overall shape of the graph.

Vertical Shift

A vertical shift occurs when you add or subtract a constant outside of a function. This moves the entire graph up or down. For example, starting with the square root function $ f(x) = \sqrt{x} $, if we look at $ f(x) = 2\sqrt{x} - 2 $, the graph is shifted vertically.

Upward Shift: Adding a constant $ c $ moves the graph up by $ c $ units.
Downward Shift: Subtracting a constant $ c $ shifts the graph down by $ c $ units.

Here, $-2$ at the end means the graph moves 2 units down. Vertical shifts do not alter the shape of the graph; they only change its vertical position.

Function Graphing

The process of function graphing involves plotting a function on a coordinate plane to visualize its behavior. With basic functions like the square root function $ f(x) = \sqrt{x} $, you start plotting from the origin and observe how $ y $ changes with increasing $ x $.
To graph transformations such as $ g(x) = 2\sqrt{x+2} - 2 $, follow these steps:

First, graph the basic function $ \sqrt{x} $.
Apply the horizontal shift to $ \sqrt{x+2} $, moving the graph 2 units left.
Perform a vertical stretch by multiplying the function by 2.
Finally, shift the entire graph down by 2 units due to the $-2$ at the end.

By understanding and following these steps, you can accurately graph even complex functions by utilizing basic transformations.

Problem 79

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