/*! 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 39 Sketch the graph of the function... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 2: Problem 39

Sketch the graph of the function, not by plotting points, but by starting with the graph of a standard function and applying transformations. $$y=3-\frac{1}{2}(x-1)^{2}$$

Short Answer

Expert verified
The graph is a downward-opening parabola, shifted right 1 unit, vertically compressed, and moved up 3 units.

Step by step solution

01

Identify the Base Function

The given function is \( y = 3 - \frac{1}{2}(x-1)^{2} \). The base function is\( y = x^2 \), which is a standard parabolic graph opening upwards.
02

Apply Horizontal Shift

The function \( (x-1)^2 \) indicates a horizontal shift. The graph of \( y = x^2 \) is shifted 1 unit to the right to get the graph of \( y = (x-1)^2 \).
03

Apply Vertical Stretch and Reflection

The expression \( -\frac{1}{2}(x-1)^2 \) implies a vertical stretch by a factor of \( \frac{1}{2} \) and a reflection over the x-axis. The graph of \( y = (x-1)^2 \) is compressed and flipped upside down to give \( y = -\frac{1}{2}(x-1)^2 \).
04

Apply Vertical Shift

Finally, the function \( y = 3 - \frac{1}{2}(x-1)^2 \) involves a vertical shift upwards by 3 units. Shift the graph of \( y = -\frac{1}{2}(x-1)^2 \) up by 3 units to get the final graph of the function.

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.

Quadratic Functions
In the realm of graph transformations, quadratic functions play a pivotal role. The standard quadratic function, represented by the equation \( y = x^2 \), outlines the simplest parabolic graph. This graph is characterized by its symmetrical bell shape, known as a parabola, that opens upwards with its vertex at the origin \((0, 0)\).
The basic properties of a quadratic function include:
  • Its graph is a parabola.
  • The direction of the opening is upwards when the coefficient of \( x^2 \) is positive.
  • The vertex represents either the maximum or minimum point of the parabola.
  • The axis of symmetry is a vertical line that intersects the vertex.
Understanding quadratic functions is fundamental, as they form the basis for more complex transformations involving shifts, stretches, and reflections. By mastering these properties, one can easily predict how a quadratic equation will transform when applying various modifications.
Horizontal Shift
A key transformation in modifying quadratic functions is the horizontal shift. This shift involves moving the entire graph sideways along the x-axis without altering its shape. Mathematically, if you see an expression of the form \((x - h)^2\), it's a sign of a horizontal shift. The parameter \( h \) determines the direction and distance of this shift.
  • If \( h \) is positive, the graph shifts to the right.
  • If \( h \) is negative, the shift is to the left.
In the function \( y = 3 - \frac{1}{2}(x-1)^{2} \), we recognize a horizontal shift due to the \((x-1)^2\) term. Here, \( h = 1 \), indicating that the graph of \( y = x^2 \) has been shifted 1 unit to the right. Horizontal shifts are visually straightforward and a critical step in graph transformations.
Vertical Stretch and Reflection
After shifting a graph horizontally, further transformations can include vertical stretches and reflections. These modifications adjust the height and orientation of the graph without affecting its horizontal position.
In the quadratic equation \( y = 3 - \frac{1}{2}(x-1)^2 \), the coefficient \(-\frac{1}{2}\) before \((x-1)^2\) plays a dual role:
  • The negative sign indicates a reflection over the x-axis, meaning the graph of the parabola flips upside down.
  • The fraction \( \frac{1}{2} \) denotes a vertical stretch of factor \( \frac{1}{2} \), which also means a compression since it's less than 1. This makes the parabola appear wider compared to the graph of \( y = x^2 \).
Understanding these transformations is important for accurately predicting how a function's graph will look, particularly when combined with other transformations like shifts.
Vertical Shift
Finally, vertical shifts play an essential role in positioning the graph vertically along the y-axis. This transformation involves moving the graph up or down without altering its shape. When looking at the equation \( y = a(x - h)^2 + k \), the constant \( k \) signals a vertical shift.
  • If \( k \) is positive, move the graph upwards.
  • If \( k \) is negative, shift the graph downwards.
In the provided function \( y = 3 - \frac{1}{2}(x-1)^2 \), the \(+3\) portion corresponds to a vertical shift upwards by 3 units. It elevates the entire graph, meaning the vertex of the parabola will no longer be at the origin but at \((1, 3)\).
Vertical shifts are often the final step in graph transformations, setting the graph precisely in its intended vertical position.

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

When a foreign object that is lodged in the trachea (windpipe) forces a person to cough, the diaphragm thrusts upward, causing an increase in pressure in the lungs. At the same time, the trachea contracts, causing the expelled air to move faster and increasing the pressure on the foreign object. According to a mathematical model of coughing, the velocity \(v\) of the airstream through an average-sized person's trachea is related to the radius \(r\) of the trachea (in centimeters) by the function $$v(r)=3.2(1-r) r^{2} \quad \frac{1}{2} \leq r \leq 1$$ Determine the value of \(r\) for which \(v\) is a maximum.

In a certain country, income tax \(T\) is assessed according to the following function of income \(x\) : $$T(x)=\left\\{\begin{array}{ll} 0 & \text { if } 0 \leq x \leq 10,000 \\ 0.08 x & \text { if } 10,000

Complete the table. $$g(x)=|2 x+3|$$ $$\begin{array}{|r|r|} \hline x & g(x) \\ \hline-3 & \\ -2 & \\ 0 & \\ 1 & \\ 3 & \\ \hline \end{array}$$

A function is given. (a) Find all the local maximum and minimum values of the function and the value of \(x\) at which each occurs. State each answer correct to two decimal places. (b) Find the intervals on which the function is increasing and on which the function is decreasing. State each answer correct to two decimal places. $$g(x)=x^{4}-2 x^{3}-11 x^{2}$$

Find the domain of the function. $$f(x)=2 x$$
See all solutions

Recommended explanations on Math Textbooks

Calculus

Read Explanation

Statistics

Read Explanation

Probability and Statistics

Read Explanation

Geometry

Read Explanation

Pure Maths

Read Explanation

Discrete Mathematics

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.