/*! 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 43 33鈥�48 ? Sketch the graph of th... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 3: Problem 43

33鈥�48 ? Sketch the graph of the function, not by plotting points, but by starting with the graph of a standard function and applying transformations. $$ y=5+(x+3)^{2} $$

Short Answer

Expert verified
Shift the parabola y=x^2 left 3 units, then up 5 units. Vertex is at (-3, 5).

Step by step solution

01

Identify the Standard Function

The given function is \(y = 5 + (x + 3)^2\). The standard function we start with here is \(y = x^2\), which is the graph of a parabola opening upwards.
02

Transformation 1 - Horizontal Shift

The expression \(x+3\) inside the square indicates a horizontal shift to the left by 3 units. So, we shift the graph of \(y = x^2\) to the left by 3 units.
03

Transformation 2 - Vertical Shift

The constant term 5 outside the square indicates a vertical shift upwards by 5 units. After shifting to the left by 3 units, we shift the entire graph up by 5 units.
04

Analyze Final Graph

The final graph is \(y = (x+3)^2\) shifted up by 5 units. It retains the shape of a parabola opening upwards, with its vertex now at \((-3, 5)\).

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.

Parabola Graph
A parabola graph is a type of curve where each point is at an equal distance from a fixed point (the focus) and a fixed straight line (the directrix). For the function \( y = x^2 \), the graph is a simple upward-opening parabola with a vertex at the origin \((0,0)\).
The standard parabola \( y = x^2 \) is symmetric about the y-axis, and the vertex is the lowest point on the graph, called the minimum point.
  • The equation of a parabola can often be expressed in the form \( y = ax^2 + bx + c \), where \( a \), \( b \), and \( c \) are constants.
  • If \( a > 0 \), the parabola opens upwards; if \( a < 0 \), it opens downwards.
The graph does not change direction when zeros or intercepts are present; rather, it rises or falls away from those points depending on the value and sign of \( a \). Understanding this helps in sketching the graph accurately.
Horizontal Shift
The horizontal shift involves moving the graph of a function left or right along the x-axis. Given an expression of \( y = (x + d)^2 \), it signifies a horizontal shift. Remember:
  • If \( d > 0 \), the graph shifts to the left by \( d \) units.
  • If \( d < 0 \), the graph shifts to the right by \( |d| \) units.
In our example, \( y = 5 + (x+3)^2 \), the expression \( (x+3) \) translates to a horizontal shift of the graph of \( y = x^2 \) to the left by 3 units.
This means every point on the original graph moves 3 units leftward. It's like re-centering the entire graph slightly to the side, which impacts where the vertex鈥攏ow \((-3, 0)\)鈥攚ill be located on the x-axis, before any further transformations.
Vertical Shift
A vertical shift moves a graph up or down along the y-axis. This occurs when a constant is added or subtracted from the function. For the general function \( y = f(x) + c \):
  • If \( c > 0 \), the graph shifts upward by \( c \) units.
  • If \( c < 0 \), the graph shifts downward by \( |c| \) units.
In the function \( y = 5 + (x+3)^2 \), the constant \( +5 \) suggests a vertical shift upward by 5 units. After applying the horizontal shift, the entire graph then moves up 5 units along the y-axis.
This vertical repositioning changes the location of the vertex鈥攆rom \((-3, 0)\) after the horizontal shift, to \((-3, 5)\) after the vertical shift is applied. Each point on the graph follows suit, leading to the final positioned graph.
Vertex Form
Vertex form is a way to express the equation of a parabola, which makes it easy to determine its vertex and direction. The formula is generally written as \( y = a(x-h)^2 + k \). Here鈥檚 how you can interpret it:
  • The vertex of the parabola is given by the point \( (h, k) \).
  • The value \( a \) determines the width and direction (upward or downward) of the parabola.
In the scenario of \( y = 5 + (x+3)^2 \), we can rewrite this in vertex form as \( y = (x+3)^2 + 5 \). Here:
  • \( h = -3 \), so the vertex is horizontally shifted left to \( x = -3 \).
  • \( k = 5 \), so the vertex is vertically shifted up to \( y = 5 \).
  • Hence, the vertex is \((-3, 5)\).
Understanding the vertex form is crucial because it provides a clear picture of the parabola's positioning and orientation without having to graph it point-by-point.

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

Sums of Even and Odd Functions If \(f\) and \(g\) are both even functions, is \(f+g\) necessarily even? If both are odd, is their sum necessarily odd? What can you say about the sum if one is odd and one is even? In each case, prove your answer.

\(1-6\) Find \(f+g, f-g, f g,\) and \(f / g\) and their domains. $$ f(x)=\sqrt{9-x^{2}}, \quad g(x)=\sqrt{x^{2}-4} $$

\(17-22=\) Use \(f(x)=3 x-5\) and \(g(x)=2-x^{2}\) to evaluate the expression. $$ \begin{array}{ll}{\text { (a) }(f \circ f)(x)} & {\text { (b) }(g \circ g)(x)}\end{array} $$

Sales Growth The annual sales of a certain company can be modeled by the function \(f(t)=4+0.01 t^{2}\) , where \(t\) represents years since 1990 and \(f(t)\) is measured in millions of dollars. (a) What shifting and shrinking operations must be performed on the function \(y=t^{2}\) to obtain the function \(y=f(t) ?\) (b) Suppose you want \(t\) to represent years since 2000 instead of \(1990 .\) What transformation would you have to apply to the function \(y=f(t)\) to accomplish this? Write the new function \(y=g(t)\) that results from this transformation.

A family of functions is given. In parts (a) and (b) graph all the given members of the family in the viewing rectangle indicated. In part (c) state the conclusions you can make from your graphs. $$f(x)=c x^{2}$$ (a) \(c=1, \frac{1}{2}, 2,4 ; \quad[-5,5]\) by \([-10,10]\) (b) \(c=1,-1,-\frac{1}{2},-2 ;[-5,5]\) by \([-10,10]\) (c) How does the value of \(c\) affect the graph?
See all solutions

Recommended explanations on Math Textbooks

Theoretical and Mathematical Physics

Read Explanation

Mechanics Maths

Read Explanation

Pure Maths

Read Explanation

Calculus

Read Explanation

Logic and Functions

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.