Problem 24
Question
Explain how the following functions can be obtained from \(y=x^{3}\) by basic transformations: (a) \(y=x^{3}+1\) (b) \(y=(x+1)^{3}-1\) (c) \(y=-3(x-2)^{3}\)
Step-by-Step Solution
Verified Answer
(a) Up 1 unit, (b) Left 1 unit, down 1 unit, (c) Right 2 units, reflect, stretch.
1Step 1: Understanding Function Transformations
Transformations include translations (shifting), reflections (flipping), stretches, and compressions. Starting with the function \( y = x^3 \), our goal is to apply these transformations to obtain the given functions.
2Step 2: Translation Transformation for (a)
To transform \( y = x^3 \) into \( y = x^3 + 1 \), you'll need to apply a vertical translation. This involves shifting the graph upward by 1 unit. This transformation affects the entire graph, moving every point up by the same amount without affecting the shape of the graph.
3Step 3: Combining Horizontal Shift and Vertical Shift for (b)
In \( y = (x+1)^3 - 1 \), we have two transformations. First, \( x \) is replaced by \( x+1 \), which means the graph of \( y = x^3 \) is shifted to the left by 1 unit horizontally. Next, subtracting 1 corresponds to a downward vertical shift of 1 unit. Thus, the graph is shifted left 1 unit and then down 1 unit.
4Step 4: Reflection and Stretch for (c)
The function \( y = -3(x-2)^3 \) involves a reflection and a vertical stretch. First, \( x-2 \) indicates a horizontal shift to the right by 2 units. The coefficient \(-3\) implies the graph is reflected over the x-axis (because of the negative sign) and vertically stretched by a factor of 3 (because of the 3). This means the steepness of the curve is increased.
Key Concepts
Translation TransformationHorizontal ShiftVertical Stretch
Translation Transformation
Translation transformation involves shifting the entire graph of a function without changing its shape or orientation. It is one of the simplest types of transformations. When we translate a graph, all points move in the same direction and by the same amount.
In case (a) from the exercise, where the function changes from \( y = x^3 \) to \( y = x^3 + 1 \), a vertical translation occurs. This means the entire graph moves upward by 1 unit. Every point on the original graph is displaced the same amount vertically. The essential characteristics of the cubic function, such as its symmetry and curvature, remain unchanged.
Translations can be:
In case (a) from the exercise, where the function changes from \( y = x^3 \) to \( y = x^3 + 1 \), a vertical translation occurs. This means the entire graph moves upward by 1 unit. Every point on the original graph is displaced the same amount vertically. The essential characteristics of the cubic function, such as its symmetry and curvature, remain unchanged.
Translations can be:
- Vertical: Adding or subtracting a constant from the function. E.g., adding 1 results in moving the graph up by 1 unit.
- Horizontal: Shifting the graph left or right by altering the input variable, essentially changing the function’s domain.
Horizontal Shift
A horizontal shift is another type of transformation where the graph moves to the left or the right along the x-axis. This type of shift involves changing the input variable's location within the function, making it more subtle than vertical shifts.
For example, in exercise (b), the function changes from \( y = x^3 \) to \( y = (x+1)^3 - 1 \). Here, \( x \) is replaced by \( x+1 \). This replacement signifies that every point on the graph is shifted left by 1 unit. The function itself encloses \( x+1 \), indicating leftward movement, occurring in the opposite direction of the sign inside the bracket.
Horizontal shifts work in the following ways:
For example, in exercise (b), the function changes from \( y = x^3 \) to \( y = (x+1)^3 - 1 \). Here, \( x \) is replaced by \( x+1 \). This replacement signifies that every point on the graph is shifted left by 1 unit. The function itself encloses \( x+1 \), indicating leftward movement, occurring in the opposite direction of the sign inside the bracket.
Horizontal shifts work in the following ways:
- Left Shift: Occurs when a positive number is added to x \((x + n)\), moving the graph left by "n" units.
- Right Shift: Happens when a negative number is added \((x - n)\), moving the graph right by "n" units.
Vertical Stretch
Vertical stretches adjust the output values of the function, affecting the "tallness" or "flatness" of the graph. Significantly, these transformations multiply the function, altering the graph's steepness without changing horizontal placements.
In scenario (c), where \( y = x^3 \) transforms into \( y = -3(x-2)^3 \), there’s a vertical stretching by a factor of 3. This multiplication increases the steepness, making the graph vertically longer. The slope of any tangent at any point becomes steeper compared to the original function. If the coefficient were between 0 and 1, it would result in a vertical compression, making the graph appear flatter.
Vertical stretches reflect some unique features:
In scenario (c), where \( y = x^3 \) transforms into \( y = -3(x-2)^3 \), there’s a vertical stretching by a factor of 3. This multiplication increases the steepness, making the graph vertically longer. The slope of any tangent at any point becomes steeper compared to the original function. If the coefficient were between 0 and 1, it would result in a vertical compression, making the graph appear flatter.
Vertical stretches reflect some unique features:
- Greater than 1: Multiplying by values greater than 1 causes vertical stretches.
- Negative Coefficient: The graph reflects around the x-axis in addition to being vertically stretched when multiplied by a negative.
Other exercises in this chapter
Problem 23
Graph \(y=x^{n}, x \geq 0\), for \(n=1,2,3\), and 4 in one coordinate system. Where do the curves intersect?
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(a) Graph \(f(x)=x, x \geq 0\), and \(g(x)=x^{2}, x \geq 0\), together, in one coordinate system. (b) For which values of \(x\) is \(f(x) \geq g(x)\), and for w
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Determine the equation of the line that satisfies the stated requirements. Put the equation in standard form. The line with slope 1 and \(x\) -intercept \((-2,0
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