Problem 7 Write a function $g$ whose gra... [FREE SOLUTION]

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BIG IDEAS MATH Algebra 2: Common Core Student Edition 2015

Ron Larson, Laurie Boswell

$Math Studyset 91影视 Explanations$ Math

1 Edition

Chapter 1: Problem 7

Write a function $g$ whose graph represents the indicated transformation of the graph of $f$. Use a graphing calculator to check your answer. $f(x)=4-|x+1|$

Short Answer

Expert verified

The function $g$ whose graph represents the indicated transformation of the function $f$ is $g(x) = 4 - |x + 1|$

Step by step solution

Identify the parent function and its behavior

The given function is $f(x) = 4 - |x+1|$. By analyzing this function, the parent function can be found as $f(x) = |x|$. The parent function $|x|$ has a V-shape graph with the vertex at the origin (0,0).

Determine the transformation effects by analyzing given function

In function $f(x)$, $4 - |x+1|$, the '-1' inside the absolute value function will cause the graph to shift to the left by 1 unit. The 4 outside the absolute value function means the graph shifts upwards by 4 units. Finally, the negative sign in front of the absolute value means the graph inverts about the x-axis. So, the transformation is left shift by 1 unit, upwards shift by 4 units, and reflection over the x-axis.

Write the new function

Since the transformation causes a left shift, upward shift, and reflection, the transformed function representing this is $g(x) = 4 - |x + 1|$. This is the function whose graph represents the indicated transformation of the graph of $f$.

Graphical validation

Use a graphing calculator to plot both $f(x) = 4 - |x + 1|$ and $g(x)$. Both graphs should match perfectly, validating the correctness of the transformation function derived.

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Key Concepts

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

Absolute Value Function

The absolute value function is a mathematical expression represented as $f(x) = |x|$. This function is named for its ability to convert any input number into a non-negative number, essentially measuring its distance from zero on a number line. The graph of the absolute value function appears as a "V" shape that intersects at the origin (0,0). The characteristic sharp "V" form results from its piecewise definition:

For $x \geq 0$, $f(x) = x$
For $x < 0$, $f(x) = -x$

When you apply transformations, like horizontal or vertical shifts, or reflections, the shape of the "V" remains but its position or orientation changes. Understanding these transformations helps sketch the graph quickly from its basic shape.

Graphing Calculator

A graphing calculator is a powerful tool that assists in visualizing mathematical functions. For the given exercise, it's essential to verify transformations by plotting the functions. By inputting $g(x) = 4 - |x+1|$, you can observe the graph's behavior and transformations. Here are some steps to guide you:

Enter the function into the calculator.
Adjust the window settings to ensure you capture the graph properly.
Plot the function to see changes based on the transformations.

Utilizing a graphing calculator aligns visual confirmation with algebraic analysis. It develops your understanding by showing the exact movement of graphs after applying shifts or reflections.

Reflection Over the X-Axis

Reflection over the x-axis is a transformation where a graph is "flipped" vertically. For the function given, $f(x) = 4 - |x+1|$, the negative sign before the absolute value indicates this reflection. In essence:

Positive values become negative.
The graph maintains its shape but is inverted.

For example, if $f(x) = |x|$ has its vertex at (0,0), reflecting it over the x-axis changes it to $-|x|$. This reflection is part of creating the complete transformation for $f(x)$ in the problem. It visually confirms that distances from the x-axis are equivalent but appear below the axis.

Horizontal Shift

The horizontal shift is a critical function transformation that changes the positioning of a graph along the x-axis. In the function $f(x) = 4 - |x+1|$, you see the term $x+1$ within the absolute value. This indicates a horizontal shift:

A shift to the left by 1 unit because it is written as $+1$.

This transformation contrasts the normal expectation because the expression $x+c$ (where $c>0$) shifts a graph left, despite what might feel intuitive. This horizontal shift alters the position of the vertex of $|x|$ from the origin to (-1,0) before applying other transformations. Pairing horizontal shifts with vertical shifts and reflections paints the full graph picture for any transformed function.

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91影视

Write a function \(g\) whose graph represents the indicated transformation of the graph of \(f\). Use a graphing calculator to check your answer. \(f(x)=4-|x+1|\)

Short Answer

Step by step solution

Identify the parent function and its behavior

Determine the transformation effects by analyzing given function

Write the new function

Graphical validation

Key Concepts

Absolute Value Function

Graphing Calculator

Reflection Over the X-Axis

Horizontal Shift

One App. One Place for Learning.

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