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Chapter 7: Problem 13

(Graphing program optional.) Given \(f(x)=4 x^{2}\), construct a function that is a reflection of \(f(x)\) across the horizontal axis. Graph the functions and confirm your answer.

Short Answer

Expert verified
The reflected function is -4x^2.

Step by step solution

01

Understand the Reflection Concept

To reflect a function across the horizontal axis, multiply the function by -1. This changes the sign of all the function's output values.
02

Apply the Reflection

Given the function f(x) = 4x^2, the reflected function across the horizontal axis will be -f(x). Therefore, the new function is -f(x) = -4x^2.
03

Construct the Functions

We now have the original function f(x) = 4x^2 and the reflected function -f(x) = -4x^2.
04

Graph Both Functions

Plot the graphs of f(x) = 4x^2 and -f(x) = -4x^2. The graph of f(x) = 4x^2 is a parabola opening upwards, while the graph of -f(x) = -4x^2 is a parabola opening downwards. Confirm that the graphs are reflections of each other across the horizontal axis.

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Key Concepts

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

graphing functions
Graphing functions is a key skill in algebra and calculus. When you graph a function, you draw the curve that represents the function's equation. For instance, for a function like \(f(x) = 4x^2\), you plot points for various values of \(x\), then draw a smooth curve through those points.

To do this:
  • Choose several values for \(x\).
  • Calculate the corresponding \(y\) values by plugging \(x\) into the function.
  • Plot these \((x,y)\) points on a graph.
For \(f(x) = 4x^2\), this creates a parabola that opens upwards because the coefficient of \(x^2\) is positive. When you graph its reflection \(-f(x)\), the parabola opens downwards, as the coefficient of \(x^2\) becomes negative. Observing the graph can help you understand function properties and transformations better.
quadratic functions
A quadratic function is a type of polynomial function that forms a parabola when graphed. It has the standard form \(ax^2 + bx + c\), where \(a\), \(b\), and \(c\) are constants. The function given in the exercise, \(f(x) = 4x^2\), is a simple quadratic equation with \(b = 0\) and \(c = 0\).

Key features of quadratic functions include:
  • The vertex: The highest or lowest point on the graph.
  • The axis of symmetry: A vertical line that passes through the vertex and divides the parabola into two mirror images.
  • The direction of the parabola: Upward if \(a > 0\), and downward if \(a < 0\).
Quadratics are significant in various fields such as physics, engineering, and economics, due to their distinct properties and ease of calculation.
transformations in algebra
Transformations in algebra include translations, reflections, stretches, and compressions. These transformations are operations that alter the graph of a function in different ways:
  • Translation: Moves the graph horizontally or vertically without changing its shape.
  • Reflection: Flips the graph over a specified axis. In this exercise, reflecting \(f(x)\) across the horizontal axis gives \(-f(x)\).
  • Stretch/Compression: Alter the graph's width or height. For instance, multiplying \(f(x)\) by a factor greater than 1 stretches it, while a factor between 0 and 1 compresses it.
Understanding these transformations is essential in algebra because they help us graph functions quickly and understand their behaviors. In the exercise, reflecting the function \(f(x) = 4x^2\) creates a new function \(-f(x) = -4x^2\), demonstrating how the graph of functions can be manipulated through algebraic transformations.

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Most popular questions from this chapter

The rate of vibration of a string under constant tension is inversely proportional to the length of the string. a. Write an equation for the vibration rate of a string, \(v,\) as a function of its length, \(l\). b. If a 48 -inch string vibrates 256 times per second, then how long is a string that vibrates 512 times per second? c. In general, it can be said that if the length of the string increases, the vibration rate will d. If you want the vibration rate of a string to increase, then you must the length of the string. e. Playing a stringed instrument, such as a guitar, dulcimer, banjo, or fiddle, requires placing your finger on a fret, effectively shortening the string. Doubling the vibration produces a note pitched one octave higher, and halving the vibration produces a note pitched one octave lower. If the number of vibrations decreased from 440 to 220 vibrations per second, what happened to the length of the string to cause the change in vibration?

(Graphing program recommended.) Graph \(y=x^{4}\) and \(y=4^{x}\) on the same grid. a. For positive values of \(x\), where do your graphs intersect? Do they intersect more than once? b. For positive values of \(x\), describe what happens to the right and left of any intersection points. You may need to change the scales on the axes or change the windows on a graphing calculator in order to see what is happening. c. Which eventually dominates, \(y=x^{4}\) or \(y=4^{x} ?\)

(Graphing program optional.) Evaluate each of the following functions at \(0,0.5,\) and \(1 .\) Then, on the same grid, graph each over the interval [0,1] . Compare the graphs. a. \(y_{1}=x \quad y_{2}=x^{1 / 2} \quad y_{3}=x^{1 / 3} \quad y_{4}=x^{1 / 4}\) b. \(y_{5}=x^{2} \quad y_{6}=x^{3} \quad y_{7}=x^{4}\)

Assume \(Y\) is directly proportional to \(X^{3}\). a. Express this relationship as a function where \(Y\) is the dependent variable. b. If \(Y=10\) when \(X=2\), then find the value of the constant of proportionality in part (a). c. If \(X\) is increased by a factor of 5 , what happens to the value of \(Y ?\) d. If \(X\) is divided by \(2,\) what happens to the value of \(Y ?\) e. Rewrite your equation from part (a), solving for \(X .\) Is \(X\) directly proportional to \(Y ?\)

a. An insulation blanket for a cylindrical hot water heater is sold in a roll 48 in \(\times 75\) in \(\times 2\) in. Assuming a hot water heater 48 inches high, for what diameter water heater is this insulation blanket made? (Round to the nearest \(\frac{1}{2}\) inch.) b. What is the volume of the hot water heater in part (a)? ( Note: Ignore the thickness of the heater walls.) c. If 1 gallon \(=231\) in \(^{3},\) what is the maximum number of gallons of water that a cylinder the size of this water heater could hold?
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