Problem 14 Sketch, on the same coordinate p... [FREE SOLUTION]

Chapter 2: Problem 14

Sketch, on the same coordinate plane, the graphs of $f$ for the given values of $c$. (Make use of symmetry, shifting, stretching, compressing, or reflecting.) $$f(x)=|x-c| ; \quad \quad c=-3,1,3$$

Short Answer

Expert verified

The graphs for $ f(x) = |x-c| $ with $ c = -3, 1, 3 $ are 'V' shapes shifted left for $ c = -3 $, right for $ c = 1 $ and $ c = 3 $.

Step by step solution

Understanding the Function

The function given is $ f(x) = |x - c| $. This is an absolute value function, which typically forms a 'V' shape on the graph.

Analyzing Parameter 'c'

The value of $ c $ affects the horizontal shift of the graph. For each different value of $ c $, the graph undergoes a horizontal shift to the right (if $ c $ is positive) or to the left (if $ c $ is negative) by $ |c| $ units.

Plotting for $ c = -3 $

For $ c = -3 $, the function becomes $ f(x) = |x + 3| $. This shifts the standard graph $ |x| $ 3 units to the left. The vertex of the 'V' is at $ (-3, 0) $.

Plotting for $ c = 1 $

For $ c = 1 $, the function is $ f(x) = |x - 1| $. This shifts $ |x| $ 1 unit to the right. The vertex of the graph is at $ (1, 0) $.

Plotting for $ c = 3 $

For $ c = 3 $, the function is $ f(x) = |x - 3| $. This shifts the graph of $ |x| $ 3 units to the right. The vertex of the 'V' is at $ (3, 0) $.

Sketching the Graphs

On the coordinate plane, plot the vertices at $(-3, 0)$, $(1, 0)$, and $(3, 0)$ for the respective functions. Each graph retains the 'V' shape and all open upwards, demonstrating their respective shifts.

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Key Concepts

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

Graph Transformations

Graph transformations involve changing the position or shape of a graph by applying specific rules. For absolute value functions, the general formula is $ f(x) = |x-c| $. This formula represents a standard absolute value function, which has the shape of a "V".

Key transformations for absolute value functions include:

Horizontal Shifts: Adjustments along the x-axis produced by changing the value of $ c $ in $ |x-c| $.
Vertical Shifts: Movements along the y-axis, though not part of this problem, occur if you alter the expression to something like $ |x| + d $.
Reflections: Flipping the graph over the x-axis, which would involve negating the whole function like $-|x-c|$.
Stretching and Compressing: Scaling the graph, which is done by multiplying the function by a constant, such as $ a|x-c| $ where $ a eq 1 $.

By using these transformations, you can turn a simple "V" shape into a modified version that fits the requirements of any particular equation. This allows flexibility in graphing complex functions by breaking them down into simpler transformations.

Horizontal Shifts

Horizontal shifts in graph transformations occur when the entire graph moves left or right along the x-axis. For the absolute value function $ f(x) = |x-c| $, the parameter $ c $ dictates how far and in which direction the graph shifts.

Here's how it works:

If $ c $ is positive, the graph shifts to the right by $ c $ units.
If $ c $ is negative, the graph shifts to the left by $ |c| $ units.

For the given function with different values of $ c $:

$ c = -3 $: The graph of $|x|$ shifts 3 units to the left, resulting in $ f(x) = |x+3| $ with a vertex at $(-3, 0)$.
$ c = 1 $: The graph shifts 1 unit to the right, resulting in $ f(x) = |x-1| $ with a vertex at $(1, 0)$.
$ c = 3 $: The graph shifts 3 units to the right, resulting in $ f(x) = |x-3| $ with a vertex at $(3, 0)$.

These shifts indicate the new position of the vertex - the point where the graph changes direction.

Vertex of Absolute Value Function

The vertex of an absolute value function is the crucial point where the graph changes direction, forming a "V" shape at this pivot. In the function $ f(x) = |x-c| $, the vertex is located at the point $(c, 0)$. This is because the expression inside the absolute value becomes zero at $ x = c $, positioning the lowest point of the graph (if it opens upwards) or the highest point (if it opens downwards, such as in $ f(x) = -|x-c| $).

The vertex offers important information:

The location of the vertex directly tells us about horizontal shifts.
Knowing the vertex helps in quickly sketching the graph since it dictates the direction and scale of the function.

For the absolute value functions in the problem:

The function $ f(x) = |x+3| $ has its vertex at $(-3, 0)$.
The function $ f(x) = |x-1| $ has its vertex at $(1, 0)$.
The function $ f(x) = |x-3| $ has its vertex at $(3, 0)$.

These vertices provide critical reference points for graphing and interpreting the behavior of absolute value functions in various contexts.

91影视

Sketch, on the same coordinate plane, the graphs of \(f\) for the given values of \(c\). (Make use of symmetry, shifting, stretching, compressing, or reflecting.) $$f(x)=|x-c| ; \quad \quad c=-3,1,3$$

Short Answer

Step by step solution

Understanding the Function

Analyzing Parameter 'c'

Plotting for \( c = -3 \)

Plotting for \( c = 1 \)

Plotting for \( c = 3 \)

Sketching the Graphs

Key Concepts

Graph Transformations

Horizontal Shifts

Vertex of Absolute Value Function

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