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Chapter 0: Problem 51

Use transformations to graph each function. $$ y=-\frac{1}{2}|x+4| $$

Short Answer

Expert verified
Shift the graph of \( y = |x| \) left by 4 units, compress it vertically by \( \frac{1}{2} \), and finally reflect it across the x-axis.

Step by step solution

01

Understand the Parent Function

The parent function to this equation is the absolute value function, given by \( y = |x| \). This is a V-shaped graph that opens upwards and has its vertex at the origin (0,0).
02

Horizontal Shift

The equation \( y = -\frac{1}{2}|x+4| \) indicates a horizontal shift. The term \( x+4 \) shifts the graph to the left by 4 units. So the vertex of the transformed graph will be at \((-4,0)\).
03

Vertical Compression

The coefficient \(-\frac{1}{2}\) affects the steepness of the graph. The absolute value of \(-\frac{1}{2}\) which is \( \frac{1}{2} \) compresses the graph vertically by a factor of \( \frac{1}{2} \), meaning it will be half as steep as the parent function.
04

Reflection

The negative sign in front of the fraction \( -\frac{1}{2} \) reflects the graph across the x-axis. This will make the V-shape open downwards instead of upwards.
05

Combine Transformations

Now, combine all the transformations: Start with the parent graph \( y = |x| \), shift it left by 4 units to \( y = |x+4| \), compress it vertically by a factor of \( \frac{1}{2} \) to \( y = \frac{1}{2}|x+4| \), and lastly reflect it across the x-axis to obtain \( y = -\frac{1}{2}|x+4| \).
06

Graph the Function

Draw the V-shaped graph starting at the vertex \((-4, 0)\), opening downwards, and being half as steep as the original absolute value function. This completes the transformation.

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Key Concepts

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

Parent Function
The parent function of the given equation is the absolute value function, represented as \( y = |x| \). This function forms a V-shaped graph that opens upwards and has its vertex at the origin, (0,0).
Understanding the parent function is crucial because all transformations will be applied to this base function.
In our exercise, our job is to transform this parent function to get the final graph.
Horizontal Shift
Next, we have the horizontal shift, which moves the graph left or right.
In the equation \( y = -\frac{1}{2}|x+4| \), the term \( x+4 \) shows a horizontal shift.
When a function has \( x + c \) inside, it shifts left by \( c \) units.
So, \( x+4 \) moves our parent function 4 units to the left.
The new vertex of the graph will be at (-4, 0).
Vertical Compression
The next transformation involves vertical compression.
In our equation, the coefficient ' \( -\frac{1}{2} \) ' affects this.
Normally, the absolute value function has a slope of 1 or -1.
Here, the absolute value of \( -\frac{1}{2} \) which is \( \frac{1}{2} \) compresses the graph vertically by a factor of \( \frac{1}{2} \).
This means the graph will be half as steep as the parent function, making it 'wider'.
Reflection
The final transformation is reflection.
The negative sign in front of \( -\frac{1}{2} \) reflects the graph across the x-axis.
While the parent function opens upwards, after reflection it will open downwards.
This changes the direction of the 'V' shape.

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