Problem 26
Question
Explain how the following functions can be obtained from \(y=1 / x^{2}\) by basic transformations: (a) \(y=\frac{1}{x^{2}}+1\) (b) \(y=-\frac{1}{(x+1)^{2}}\) (c) \(y=-\frac{1}{x^{2}}-2\)
Step-by-Step Solution
Verified Answer
(a) Vertical shift up by 1; (b) Left shift 1 unit, reflection over x-axis; (c) Reflection over x-axis, vertical shift down by 2.
1Step 1: Identify the Base Function
The base function given is \( y = \frac{1}{x^2} \). We will apply transformations to this base function to obtain each of the desired functions.
2Step 2: Transformation to Obtain (a)
For \( y = \frac{1}{x^2} + 1 \), compare it with the base function. This can be obtained by adding 1 to the base function. This is a vertical shift upwards by 1 unit of the basic curve \( y = \frac{1}{x^2} \).
3Step 3: Transformation to Obtain (b)
For \( y = -\frac{1}{(x+1)^2} \), first note the negative sign and the \((x+1)\) term. The negative sign indicates reflection about the x-axis. Replacing \(x\) with \(x+1\) shifts the graph 1 unit to the left. Overall, it is a left shift by 1 unit followed by a reflection across the x-axis of the original function.
4Step 4: Transformation to Obtain (c)
For \( y = -\frac{1}{x^2} - 2 \), the negative sign implies reflection about the x-axis. The subtraction of 2 indicates a vertical shift downwards by 2 units. Therefore, it is reflection and then a downward shift of 2 units of the base function \( y = \frac{1}{x^2} \).
Key Concepts
Vertical ShiftReflection Across x-axisHorizontal Shift
Vertical Shift
In function transformations, a vertical shift is a movement of the graph of a function up or down on the coordinate plane. To achieve this, we either add or subtract a constant from the function.
For instance, if we start with a base function like \( y = \frac{1}{x^2} \) and consider the transformation \( y = \frac{1}{x^2} + 1 \), we see a vertical shift upwards by 1 unit. Here, the entire graph of the function moves one unit up.
This upward shift affects each point on the graph equally without changing the shape of the graph. The vertical shift only changes the position of the entire graph. Consider another transformation such as \( y = \frac{1}{x^2} - 2 \). This represents a shift downward by 2 units.
So, to summarize:
For instance, if we start with a base function like \( y = \frac{1}{x^2} \) and consider the transformation \( y = \frac{1}{x^2} + 1 \), we see a vertical shift upwards by 1 unit. Here, the entire graph of the function moves one unit up.
This upward shift affects each point on the graph equally without changing the shape of the graph. The vertical shift only changes the position of the entire graph. Consider another transformation such as \( y = \frac{1}{x^2} - 2 \). This represents a shift downward by 2 units.
So, to summarize:
- Addition of a constant \( k \) to the function results in an upward shift of \( k \) units, \( y = f(x) + k \).
- Subtraction of a constant \( k \) results in a downward shift of \( k \) units, \( y = f(x) - k \).
Reflection Across x-axis
Reflection across the x-axis is a transformation that flips the graph of a function over the x-axis. Imagine the graph being mirrored below the x-axis when this transformation is applied.
This is done by multiplying the entire function by -1. If we take the base function \( y = \frac{1}{x^2} \), a reflection across the x-axis transforms it into \( y = -\frac{1}{x^2} \). Every point on the graph is mirrored vertically – if a point was above the x-axis, it now mirrors directly below and vice versa.
When reflecting, keep in mind:
In transformations, this reflection often combines with other operations such as shifting. For example, \( y = -\frac{1}{x^2} - 2 \) involves reflecting across the x-axis and, afterwards, shifting downward by 2 units.
This is done by multiplying the entire function by -1. If we take the base function \( y = \frac{1}{x^2} \), a reflection across the x-axis transforms it into \( y = -\frac{1}{x^2} \). Every point on the graph is mirrored vertically – if a point was above the x-axis, it now mirrors directly below and vice versa.
When reflecting, keep in mind:
- Multiplying the function by -1 flips the graph, \( y = -f(x) \).
- The shape does not change, only the position relative to the x-axis does.
In transformations, this reflection often combines with other operations such as shifting. For example, \( y = -\frac{1}{x^2} - 2 \) involves reflecting across the x-axis and, afterwards, shifting downward by 2 units.
Horizontal Shift
A horizontal shift involves moving the graph of a function left or right across the coordinate plane. This type of transformation does not change the shape of the graph; it only alters its position horizontally.
You achieve a horizontal shift by adding or subtracting a constant from the variable \( x \) in the function. For an example, if we start with the base function \( y = \frac{1}{x^2} \) and transform it to \( y = \frac{1}{(x+1)^2} \), this indicates a horizontal shift to the left by 1 unit. Here, replacing \( x \) with \( x+1 \) shifts the entire graph leftward.
This can feel a bit tricky because it seems counterintuitive:
Remember, horizontal shifts do not affect the vertical position of the graph; only where the graph sits along the x-axis.
You achieve a horizontal shift by adding or subtracting a constant from the variable \( x \) in the function. For an example, if we start with the base function \( y = \frac{1}{x^2} \) and transform it to \( y = \frac{1}{(x+1)^2} \), this indicates a horizontal shift to the left by 1 unit. Here, replacing \( x \) with \( x+1 \) shifts the entire graph leftward.
This can feel a bit tricky because it seems counterintuitive:
- Replacing \( x \) with \( x-c \) shifts the graph \( c \) units to the right.
- Conversely, replacing \( x \) with \( x+c \) shifts the graph \( c \) units to the left.
Remember, horizontal shifts do not affect the vertical position of the graph; only where the graph sits along the x-axis.
Other exercises in this chapter
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