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Magazine Chapter 1: Problem 17
Sketch the graph of the equation by making appropriate transformations to the graph of a basic power function. Check your work with a graphing utility. (a) \(y=2(x+1)^{2}\) (b) \(y=-3(x-2)^{3}\) (c) \(y=\frac{-3}{(x+1)^{2}}\) (d) \(y=\frac{1}{(x-3)^{5}}\)
Short Answer
Expert verified
Graph transformations involve shifts, stretches, and reflections per described steps, validated by a graphing utility.
Step by step solution
01
Analyze the Basic Function of Part (a)
The basic power function here is the quadratic function, given by \( y = x^2 \). The given function is \( y = 2(x+1)^2 \).
02
Perform Horizontal Shift for Part (a)
In \( y = 2(x+1)^2 \), replace \( x \) with \( x+1 \), which indicates a horizontal shift to the left by 1 unit. The graph of \( y = x^2 \) shifts left to \( y = (x+1)^2 \).
03
Apply Vertical Stretch for Part (a)
The coefficient 2 outside the square indicates a vertical stretch by a factor of 2. The graph is now \( y = 2(x+1)^2 \).
04
Analyze the Basic Function of Part (b)
The basic power function for part (b) is \( y = x^3 \). The given function is \( y = -3(x-2)^3 \).
05
Perform Horizontal Shift for Part (b)
Replace \( x \) with \( x-2 \) for a horizontal shift to the right by 2 units. The graph of \( y = x^3 \) shifts to \( y = (x-2)^3 \).
06
Apply Reflection and Vertical Stretch for Part (b)
The negative sign and coefficient 3 indicate a reflection over the x-axis and a vertical stretch by 3. The graph is now \( y = -3(x-2)^3 \).
07
Analyze the Basic Function of Part (c)
For \( y = \frac{-3}{(x+1)^2} \), the basic function is \( y = \frac{1}{x^2} \), which is a variation of the reciprocal function.
08
Perform Horizontal Shift for Part (c)
Replace \( x \) with \( x+1 \) for a horizontal shift left by 1 unit. The function becomes \( y = \frac{1}{(x+1)^2} \).
09
Apply Reflection and Vertical Stretch for Part (c)
The numerator -3 implies a reflection over the x-axis and a vertical stretch. The graph is now \( y = \frac{-3}{(x+1)^2} \).
10
Analyze the Basic Function of Part (d)
The function is \( y = \frac{1}{(x-3)^5} \), based on \( y = \frac{1}{x^5} \). This is also a reciprocal function, extending \( \frac{1}{x} \).
11
Perform Horizontal Shift for Part (d)
Replacing \( x \) with \( x-3 \) shifts the graph to the right by 3 units. The graph is \( y = \frac{1}{(x-3)^5} \).
12
Validate Transformations Using a Graphing Utility
Use a graphing utility to verify all transformations and the final graphs for all parts (a), (b), (c), and (d). Ensure all shifts, stretches, and reflections are accurately represented.
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Key Concepts
These are the key concepts you need to understand to accurately answer the question.
Understanding Power Functions
Power functions play a crucial role in graph transformations. A power function is of the form \( y = x^n \), where \( n \) is a constant. For example, the basic quadratic function is \( y = x^2 \) and the cubic function is \( y = x^3 \). These functions serve as a foundation for various types of transformations.
Power functions are characterized by their graphs' general shapes: parabolas for quadratic functions, and more complex shapes for higher powers. Understanding these shapes is essential before making transformations.
Power functions are characterized by their graphs' general shapes: parabolas for quadratic functions, and more complex shapes for higher powers. Understanding these shapes is essential before making transformations.
- Quadratic functions \( y = x^2 \) have U-shaped graphs.
- Cubic functions \( y = x^3 \) exhibit a more complex S-shaped curve.
Exploring Vertical Stretch
A vertical stretch changes how tall or flat a graph appears. It occurs when a power function is multiplied by a constant. For instance, considering the function \( y = 2(x+1)^2 \), the '2' outside the bracket indicates a vertical stretch by a factor of 2. This makes the parabola taller.
Vertical stretching makes the output values larger or smaller, transforming the entire graph upwards or downwards while keeping the x-values unchanged.
Vertical stretching makes the output values larger or smaller, transforming the entire graph upwards or downwards while keeping the x-values unchanged.
- If the coefficient is greater than 1 (e.g., 2 or 3), the graph becomes steeper.
- If the coefficient is between 0 and 1, it actually compresses vertically, flattening the graph.
The Mechanics of Horizontal Shift
A horizontal shift adjusts the graph of a function left or right. It happens by adding or subtracting a constant inside the function's argument. For example, in \( y = 2(x+1)^2 \), the expression \( x+1 \) indicates the graph shifts left by 1 unit.
Horizontal shifts affect all x-values, moving every point on the graph horizontally:
Horizontal shifts affect all x-values, moving every point on the graph horizontally:
- Adding a positive value inside (e.g., \( x+1 \)) shifts the graph to the left.
- Subtracting a value (e.g., \( x-2 \)) shifts it to the right.
Understanding Reflection Over the X-axis
Reflection over the x-axis is a transformation that flips the graph vertically. It changes the sign of all y-values from positive to negative or vice versa.
This can be observed in a graph when there is a negative sign in front of the function. For instance, in \( y = -3(x-2)^3 \), the negative sign implies a reflection over the x-axis.
Here's how reflection over the x-axis plays out:
This can be observed in a graph when there is a negative sign in front of the function. For instance, in \( y = -3(x-2)^3 \), the negative sign implies a reflection over the x-axis.
Here's how reflection over the x-axis plays out:
- Multiplies each y-value by -1, flipping it around the x-axis.
- This results in a graph that looks like a mirror image of the original.
- The curve remains the same; only its orientation is reversed.
