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Chapter 2: Problem 49

Graph the function by applying an appropriate reflection. $$ h(x)=-x^{3} $$

Short Answer

Expert verified
Reflect the graph of y = x^3 over the x-axis to get the graph of y = -x^3.

Step by step solution

01

Identify the Basic Function

First, identify the basic function related to the given function. In this case, the basic function is the cubic function: h(x) = x^3
02

Identify the Transformation

Analyze the transformation applied to the basic function. Here, the given function is h(x) = -x^3. The minus sign in front of the cubic function indicates a reflection over the x-axis.
03

Graph the Basic Function

Begin by graphing the basic function y = x^3. This graph passes through the origin (0,0) and has points (1,1) and (-1,-1), forming a curve that goes from the third quadrant to the first quadrant.
04

Apply the Reflection

Reflect the graph of y = x^3 over the x-axis to get the graph of y = -x^3. Each point (x, y) of the graph of y = x^3 will transform to the point (x, -y) on the graph of y = -x^3. For instance, the point (1, 1) becomes (1, -1) and the point (-1, -1) becomes (-1, 1).
05

Draw the Final Graph

Plot the transformed points on the coordinate plane and smoothly connect them to complete the graph. The graph of y = -x^3 will pass through points such as (0,0), (1,-1), and (-1,1).

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Key Concepts

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

cubic function
The cubic function is a fundamental type of polynomial function. It's expressed in the form:
\( f(x) = x^3 \).
Unlike quadratic functions, which produce parabolas, cubic functions create unique S-shaped curves. These curves move from one quadrant, sweep through the origin, and continue into another quadrant. For example, the standard cubic function y = x^3 will pass through key points such as (0, 0), (1, 1), and (-1, -1). Cubic functions are essential in various mathematical analyses and can describe real-world phenomena such as population growth and financial trends.
reflection over x-axis
Reflections in graphing involve flipping the graph around a specific axis.
In our case, reflecting over the x-axis means we take the y-values of each point on the graph and multiply them by -1.
For the cubic function \( y = x^3 \), the reflection is represented as \( y = -x^3 \). This transformation can be visualized by noting how every point on the graph of the original function \( y = x^3 \) transforms. If a point on the graph \( (x, y) \) is originally \( (1, 1) \), after the reflection, it becomes \( (1, -1) \). Similarly, \( (-1, -1) \) becomes \( (-1, 1) \). This results in an inverted curve that passes through the same x-coordinates but with opposite y-coordinates.
graph transformation
Graph transformations are changes made to the graph of a function, altering its position, shape, or orientation. These transformations can include:
  • Translations (shifting the graph horizontally or vertically)
  • Reflections (flipping the graph over an axis)
  • Dilations (stretching or compressing the graph)
In our example, the transformation applied is a reflection over the x-axis.
Here's how to perform it:
  • Identify the basic function, which in this exercise is \( h(x) = x^3 \).
  • Note the transformation, reflecting the graph over the x-axis by using \( h(x) = -x^3 \).
  • Graph the basic function and apply the transformation to each point.
This method helps in visualizing and understanding how each type of transformation affects the graph of the initial function. Efficiently mastering these transformations is key to accurately graphing more complex mathematical functions.

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Most popular questions from this chapter

A website designer creates videos on how to create websites. She sells the videos in 10 -hr packages for \(\$ 40\) each. Her one-time initial cost to produce each 10 -hr video package is \(\$ 5000\) (this includes labor and the cost of computer supplies). The cost to package and ship each \(\mathrm{CD}\) is \(\$ 2.80\). a. Write a linear cost function that represents the \(\operatorname{cost} C(x)\) to produce, package, and ship \(x\) 10-hr video packages. b. Write a linear revenue function to represent the revenue \(R(x)\) for selling \(x\) 10-hr video packages. c. Evaluate \((R-C)(x)\) and interpret its meaning in the context of this problem. d. Determine the profit if the website designer produces and sells 2400 video packages in the course of one year.

A car traveling \(60 \mathrm{mph}(88 \mathrm{ft} / \mathrm{sec})\) undergoes a constant deceleration until it comes to rest approximately 9.09 sec later. The distance \(d(t)\) (in ft) that the car travels \(t\) seconds after the brakes are applied is given by \(d(t)=-4.84 t^{2}+88 t,\) where \(0 \leq t \leq 9.09 .\) (See Example 5) a. Find the difference quotient \(\frac{d(t+h)-d(t)}{h}\). Use the difference quotient to determine the average rate of speed on the following intervals for \(t\) : b. [0,2]\(\quad(\) Hint \(: t=0\) and \(h=2)\) c. [2,4]\(\quad(\) Hint \(: t=2\) and \(h=2)\) d. [4,6]\(\quad(\) Hint \(: t=4\) and \(h=2)\) e. [6,8]\(\quad(\) Hint \(: t=6\) and \(h=2)\)

Suppose that the average rate of change of a continuous function between any two points to the left of \(x=a\) is positive, and the average rate of change of the function between any two points to the right of \(x=a\) is negative. Does the function have a relative minimum or maximum at \(a\) ?

Given \(f(x)=4 \sqrt{x}\) a. Find the difference quotient (do not simplify). b. Evaluate the difference quotient for \(x=1\), and the following values of \(h: h=1, h=0.1, h=0.01,\) and \(h=0.001\). Round to 4 decimal places. c. What value does the difference quotient seem to be approaching as \(h\) gets close to \(0 ?\)

A function is given. (See Examples \(4-5)\) a. Find \(f(x+h)\). b. Find \(\frac{f(x+h)-f(x)}{h}\). $$f(x)=x^{2}+4 x$$
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