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Chapter 2: Problem 34

Give the equation of each function whose graph is described. The graph of \(y=x^{3}\) is vertically stretched by applying a factor of \(3 .\) This graph is then reflected across the \(x\) -axis. Finally, the graph is shifted 8 units upward.

Short Answer

Expert verified
The equation is \(y = -3x^3 + 8\).

Step by step solution

01

Understanding Vertical Stretching

The original function is \(y = x^3\). A vertical stretch by a factor of 3 will multiply the \(y\)-values by 3, resulting in the function \(y = 3x^3\).
02

Reflecting Across the x-axis

To reflect the graph across the \(x\)-axis, you multiply the entire function by -1. This changes the function to \(y = -3x^3\).
03

Shifting Upward

To shift the graph 8 units upward, you add 8 to the entire function. This results in the final function \(y = -3x^3 + 8\).

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Key Concepts

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

Vertical Stretch
A vertical stretch is a transformation that involves "stretching" the graph of a function away from the x-axis. It's like pulling a rubber band and making it taller. Suppose you have the function \( y = f(x) \). To vertically stretch this function by a factor of \( a \), you multiply the function by \( a \). For example, if \( a = 3 \), the new function becomes \( y = 3f(x) \).
  • This transformation affects the \( y\)-values of the function, increasing them by the factor \( a \).
  • The x-intercepts remain unchanged, as only the vertical distances are altered.
In the exercise, applying a vertical stretch to \( y = x^3 \) by a factor of 3 modifies the function to \( y = 3x^3 \). This increases every y-value to three times its original value before any other transformation is applied.
Reflection
Reflection in mathematics is like looking in a mirror. When reflecting a graph across the x-axis, every point on the graph is flipped over. It's as if the "bottom" becomes the "top" and vice versa. You achieve this by multiplying the entire function, \( y = f(x) \), by \(-1\), resulting in \( y = -f(x) \).
  • This transformation reverses the sign of all y-values.
  • It does not affect x-intercepts as they remain at the same location since their y-values are zero.
In the exercise, after creating the vertically stretched function \( y = 3x^3 \), we apply a reflection across the x-axis. This results in \( y = -3x^3 \), flipping the graph upside down. Every point on the original stretched graph now has its y-value multiplied by -1.
Vertical Shift
A vertical shift involves moving a graph up or down without altering its shape. To shift a function \( y = f(x) \) upward by a certain number of units, you add that number to the entire function. Conversely, to shift it downward, you subtract.
  • This kind of transformation changes the y-coordinates of every point on the graph.
  • The overall shape of the graph remains the same, only its position on the y-axis changes.
Applying this to our example, after reflecting the function to obtain \( y = -3x^3 \), we shift it 8 units upward by adding 8 to the whole function. The resulting equation is \( y = -3x^3 + 8 \). This final step positions the graph higher than where it started, effectively moving all points up by 8 units without changing their x-coordinates.

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Most popular questions from this chapter

Solve each equation or inequality graphically. $$2 x+8>-|3 x+4|$$

Based on the ordered pairs seen in each table, make a conjecture about whether the function \(f\) is even, odd, or neither even nor odd. $$\begin{array}{r|r} x & f(x) \\ \hline-3 & -1 \\ -2 & 0 \\ -1 & 1 \\ 0 & 2 \\ 1 & 3 \\ 2 & 4 \\ 3 & 5 \end{array}$$

Write the equation that results in the desired transformation. Do not use a calculator. The absolute value function, vertically shrunk by applying a factor of \(\frac{1}{3}\) and reflected across the \(y\) -axis

Sketch a graph that depicts the amount of water in a 100 -gallon tank. The tank is initially empty and then filled at a rate of 5 gallons per minute. Immediately after it is full, a pump is used to empty the tank at 2 gallons per minute.

Based on the ordered pairs seen in each table, make a conjecture about whether the function \(f\) is even, odd, or neither even nor odd. $$\begin{array}{r|r} x & f(x) \\ \hline-3 & 10 \\ -2 & 5 \\ -1 & 2 \\ 0 & 1 \\ 1 & 2 \\ 2 & 5 \\ 3 & 10 \end{array}$$
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