Problem 8
Question
Suppose the graph of \(f\) is given. Describe how the graph of each function can be obtained from the graph of \(f\). (a) \(y=-2 f(x)\) (b) \(y=-\frac{1}{2} f(x)\)
Step-by-Step Solution
Verified Answer
(a) Reflect and stretch by 2; (b) Reflect and compress by 1/2.
1Step 1: Understanding the transformation function
We need to understand what the expression \(y = -2f(x)\) means. It applies two transformations to the function \(f(x)\): a vertical stretch by a factor of 2, and a reflection across the x-axis because of the negative sign.
2Step 2: Transformation for (a)
To get the graph of \(y = -2f(x)\) from \(f(x)\):1. Reflect the entire graph of \(f(x)\) across the x-axis. Each point (x, y) on \(f\) becomes (x, -y).2. Stretch the graph vertically by a factor of 2. This means each reflected point (x, -y) becomes (x, -2y).
3Step 3: Understanding the second transformation function
For the expression \(y = -\frac{1}{2}f(x)\), we again have two transformations: a reflection across the x-axis and a vertical compression by a factor of \(\frac{1}{2}\), due to the negative and fractional multiplier respectively.
4Step 4: Transformation for (b)
To obtain the graph of \(y = -\frac{1}{2}f(x)\) from \(f(x)\):1. Reflect the graph of \(f(x)\) over the x-axis, transforming each point (x, y) to (x, -y).2. Compress the graph vertically by a factor of \(\frac{1}{2}\), so each reflected point becomes (x, -\frac{1}{2}y).
Key Concepts
Reflection across the x-axisVertical StretchVertical Compression
Reflection across the x-axis
Reflecting a graph across the x-axis is like looking at its mirror image, but along this horizontal line. When you reflect a function, each point on the graph gets flipped upside down. More specifically, if a point on the original function is at
This means that positive values of the function become negative and vice versa, while the x-coordinates remain unchanged.
Reflections are usually indicated by a negative sign in front of the function, such as \(-f(x)\).
For example, if you start with a simple graph of \(f(x)\) that increases from \(f(0) = 1\) to \(f(1) = 2\), a reflection will transform those points to \(f(0) = -1\) and \(f(1) = -2\) respectively.
Reflections across the x-axis are quite intuitive because you are literally "turning the graph upside down," making it simple to visualize as long as you keep track of the y-values.
- (x, y),
- (x, -y)
This means that positive values of the function become negative and vice versa, while the x-coordinates remain unchanged.
Reflections are usually indicated by a negative sign in front of the function, such as \(-f(x)\).
For example, if you start with a simple graph of \(f(x)\) that increases from \(f(0) = 1\) to \(f(1) = 2\), a reflection will transform those points to \(f(0) = -1\) and \(f(1) = -2\) respectively.
Reflections across the x-axis are quite intuitive because you are literally "turning the graph upside down," making it simple to visualize as long as you keep track of the y-values.
Vertical Stretch
A vertical stretch is like pulling the graph of a function away from the x-axis. This makes the graph look "narrower," since it is elongated vertically. When you stretch a graph vertically by a factor of \(k\), each y-coordinate is multiplied by \(k\).
Let's understand it with an example: if \(k = 2\), a point \((x, y)\) on the graph will move to \((x, 2y)\).
This transformation causes all y-values of the graph to increase and makes its peaks higher and valleys deeper.
A vertical stretch can be seen in the expression \(y = 2f(x)\), where \(f(x)\) is stretched by a factor of 2.
If the original function goes through \(f(1) = 3\), a vertical stretch would change this to \(f(1) = 6\).
The overall shape of the function is preserved, but it becomes taller.
Let's understand it with an example: if \(k = 2\), a point \((x, y)\) on the graph will move to \((x, 2y)\).
This transformation causes all y-values of the graph to increase and makes its peaks higher and valleys deeper.
A vertical stretch can be seen in the expression \(y = 2f(x)\), where \(f(x)\) is stretched by a factor of 2.
If the original function goes through \(f(1) = 3\), a vertical stretch would change this to \(f(1) = 6\).
The overall shape of the function is preserved, but it becomes taller.
Vertical Compression
Vertical compression is the opposite of a vertical stretch. Here, the graph of the function is squeezed toward the x-axis, making it appear "wider." Each y-coordinate is multiplied by a positive factor \(k\), where \(0 < k < 1\).
For instance, if \(k = \frac{1}{2}\), a point \((x, y)\) on the graph becomes \((x, \frac{1}{2}y)\).
This reduces the distance of each point from the x-axis, making the function's peaks lower and valleys shallower.
In the expression \(y = \frac{1}{2}f(x)\), a vertical compression by factor \(\frac{1}{2}\) is applied.
For instance, if \(k = \frac{1}{2}\), a point \((x, y)\) on the graph becomes \((x, \frac{1}{2}y)\).
This reduces the distance of each point from the x-axis, making the function's peaks lower and valleys shallower.
In the expression \(y = \frac{1}{2}f(x)\), a vertical compression by factor \(\frac{1}{2}\) is applied.
- If originally \(f(2) = 4\), then after compression \(f(2) = 2\).
Other exercises in this chapter
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Express the function (or rule) in words. $$h(x)=x^{2}+2$$
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