/*! This file is auto-generated */ .wp-block-button__link{color:#fff;background-color:#32373c;border-radius:9999px;box-shadow:none;text-decoration:none;padding:calc(.667em + 2px) calc(1.333em + 2px);font-size:1.125em}.wp-block-file__button{background:#32373c;color:#fff;text-decoration:none} Problem 23 Identify the underlying basic fu... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 2: Problem 23

Identify the underlying basic function, and use transformations of the basic function to sketch the graph of the given function. $$g(x)=-3(x+2)^{2}-4$$

Short Answer

Expert verified
The underlying basic function is \(f(x) = x^{2}\). The transformations applied to the basic function for obtaining \(g(x)=-3(x+2)^{2}-4\) are a vertical reflection, a vertical stretch by a factor of 3, a 2-unit shift to the left, and a 4-unit shift downwards.

Step by step solution

01

Identify the Basic Function

The first task is to identify the basic function that is being transformed. In this case, recognized that the function \(g(x)=-3(x+2)^{2}-4\) is a transformed version of the basic quadratic function \(f(x) = x^{2}\).
02

Identify the Transformations

We can see that our function has a negative sign in front of the \(x^{2}\) term, which means there is a vertical reflection, a multiplier of 3 to the \(x^{2}\) term, indicating a vertical stretch by a factor of 3, the term \(+2\) inside the brackets, denoting a horizontal shift to the left by 2 units, and the term \(-4\) indicating a vertical shift downwards by 4 units.
03

Apply the Transformations to the Basic Function

To sketch the graph of this function, start with the graph of the basic function \(x^{2}\) and apply the transformations. First, reflect the graph over the x-axis due to the negative sign. Then, stretch the graph vertically by a factor of 3. Now, shift the graph to the left by 2 units, and finally, shift it down by 4 units. The resulting graph will represent the function \(g(x)=-3(x+2)^{2}-4\).

Unlock Step-by-Step Solutions & Ace Your Exams!

  • Full Textbook Solutions

    Get detailed explanations and key concepts

  • Unlimited Al creation

    Al flashcards, explanations, exams and more...

  • Ads-free access

    To over 500 millions flashcards

  • Money-back guarantee

    We refund you if you fail your exam.

Over 30 million students worldwide already upgrade their learning with 91影视!

Key Concepts

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

Basic Quadratic Function
Quadratic functions are among the most essential relationships in algebra. They express relationships where a variable is squared. The basic form of a quadratic function is given by \( f(x) = x^2 \). This simple equation yields a parabolic curve that opens upwards. The graph is symmetric around the y-axis and its vertex, the lowest point for this standard equation, lies at the origin (0, 0).

When you're working with quadratic equations, the key feature to look for is the exponent of 2 on the variable, signaling you are dealing with a parabola. Altering this base form by adding, subtracting, or multiplying different numbers leads to transformations that shift, stretch, compress, or reflect the parabola across the graph.
Vertical Reflection
A vertical reflection in function transformations involves flipping the graph of the function over the x-axis. This happens when a negative sign is introduced in front of the quadratic term \( x^2 \).

In our function \( g(x) = -3(x+2)^2 - 4 \), there is a negative sign in front of the quadratic term, indicating a vertical reflection. Instead of opening upwards like the parent function \( f(x) = x^2 \), the graph opens downwards.
  • The graph maintains the same shape but is inverted around the x-axis.
  • This causes the vertex, which is typically the minimum point in \( f(x)=x^2 \), to become the maximum point in the reflected graph.
Horizontal Shift
A horizontal shift occurs when a constant is added or subtracted inside the squared term of a quadratic equation. This transformation moves the parabola left or right across the graph.

For our function \( g(x) = -3(x+2)^2 - 4 \), the \(+2\) indicates a shift. The shift will be to the left, as the general rule for horizontal shifts is that \( f(x+c) \) shifts the graph \( c \) units to the left.
  • This means that each point of the graph of the basic parabola \( x^2 \) will move 2 units to the left.
  • The vertex position changes accordingly from (0, 0) to (-2, something based on other transformations).
Vertical Stretch
Vertical stretching involves changing the steepness or "width" of the parabola by multiplying \( x^2 \) by a constant factor. This makes the parabola appear narrower.

In \( g(x) = -3(x+2)^2 - 4 \), the coefficient \(-3\) represents a significant vertical stretch. Instead of the typical rise and run of \( x^2 \), each point on the graph is pushed further from the x-axis by a factor of 3.
  • If the multiplier is greater than 1, the function is vertically stretched, meaning it gets narrower.
  • This transformation affects only the vertical dimension, not the symmetry of the parabola or the horizontal positions of its points.

One App. One Place for Learning.

All the tools & learning materials you need for study success - in one app.

Get started for free

Most popular questions from this chapter

This set of exercises will draw on the ideas presented in this section and your general math background. How many zeros, real and nonreal, does the function \(f(x)=x^{4}-1\) have? How many \(x\) -intercepts does the graph of \(f\) have?

A rectangular plot situated along a river is to be fenced in. The side of the plot bordering the river will not need fencing. The builder has 100 feet of fencing available. (a) Write an equation relating the amount of fencing material available to the lengths of the three sides of the plot that are to be fenced. (b) Use the equation in part (a) to write an expression for the width of the enclosed region in terms of its length. (c) Write an expression for the area of the plot in terms of its length. (d) Find the dimensions that will yield the maximum area.

In Exercises \(97-100,\) let \(f(t)=-t^{2}\) and \(g(x)=x^{2}-1\). Find an expression for \((g \circ g)(x),\) and give the domain of \(g \circ g\).

The height of a ball after being dropped from the roof of a 200 -foot-tall building is given by \(h(t)=-16 t^{2}+200,\) where \(t\) is the time in seconds since the ball was dropped, and \(h(t)\) is in feet. (a) When will the ball be 100 feet above the ground? (b) When will the ball reach the ground? (c) For what values of \(t\) does this problem make sense (from a physical standpoint)?

A ball is thrown directly upward from ground level at time \(t=0\) ( \(t\) is in seconds). At \(t=3,\) the ball reaches its maximum distance from the ground, which is 144 feet. Assume that the distance of the ball from the ground (in feet) at time \(t\) is given by a quadratic function \(d(t) .\) Find an expression for \(d(t)\) in the form \(d(t)=a(t-h)^{2}+k\) by performing the following steps. (a) From the given information, find the values of \(h\) and \(k\) and substitute them into the expression \(d(t)=a(t-h)^{2}+k\) (b) Now find \(a\). To do this, use the fact that at time \(t=0\) the ball is at ground level. This will give you an equation having just \(a\) as a variable. Solve for \(a\) (c) Now, substitute the value you found for \(a\) into the expression you found in part (a). (d) Check your answer. Is (3,144) the vertex of the associated parabola? Does the parabola pass through (0,0)\(?\)
See all solutions

Recommended explanations on Math Textbooks

Calculus

Read Explanation

Decision Maths

Read Explanation

Probability and Statistics

Read Explanation

Pure Maths

Read Explanation

Mechanics Maths

Read Explanation

Logic and Functions

Read Explanation
View all explanations

What do you think about this solution?

We value your feedback to improve our textbook solutions.

Study anywhere. Anytime. Across all devices.