Problem 94 Begin by graphing the absolute v... [FREE SOLUTION]

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Precalculus Essentials

Robert Blitzer

$Math Studyset 91影视 Explanations$ Math

5 Edition

Chapter 1: Problem 94

Begin by graphing the absolute value function, $f(x)=|x| .$ Then use transformations of this graph to graph the given function. $$g(x)=-2|x+3|+2$$

Short Answer

Expert verified

The process to graph the function $g(x)=-2|x+3|+2$ includes three transformations. Start with the base graph of $f(x) = |x|$, then apply a horizontal shift of 3 units to the left for the $|x+3|$ portion, reflect over the x-axis due to the '-2' outside the absolute value sign, and finally move the graph upwards by 2 units resulting from '+2' outside the absolute value sign.

Step by step solution

Graph the Absolute Value Function

First begin by drawing the graph for $f(x) = |x|$. This is a V-shaped graph which starts at the origin (0,0). The graph increases by a slope of 1 for all $x \geq 0$ and decreases by a slope of -1 for all $x < 0$.

Understanding the Transformations

For the function $g(x)=-2|x+3|+2$, there are three transformations to apply to the base absolute value function: \nThe graph is translated 3 units to the left, because of the '+3' inside the absolute value brackets. This means every $x$ in $f(x)$ will be replaced by $(x+3)$. This is a horizontal shift. \nThe graph is reflected over the x-axis, because of the '鈭�2' coefficient of the absolute value. This means we multiply the whole function $f(x)$ by -2, changing the direction of the V of the graph. \nLastly, the graph is translated 2 units up, due to '+2' at the end of the equation. This is a vertical shift, meaning the entire graph of $f(x)$ is moved up by 2 units.

Graphing the Function

Begin by moving every point in the base absolute value function 3 units to the left, altering if $x$ is greater or smaller than 0. Then reflect the graph over the x-axis, changing the direction that the V opens. Lastly, move each point 2 units upwards. The graph that you draw should be the one ending up with the equation $g(x) = -2|x+3| + 2$.

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91影视

Begin by graphing the absolute value function, \(f(x)=|x| .\) Then use transformations of this graph to graph the given function. $$g(x)=-2|x+3|+2$$

Short Answer

Step by step solution

Graph the Absolute Value Function

Understanding the Transformations

Graphing the Function

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