/*! 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 7 In Exercises \(5-12,\) sketch th... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 1: Problem 7

In Exercises \(5-12,\) sketch the graph of the equation by translating, reflecting, compressing. and stretching the graph of \(y=x^{2}\) appropriately. and then use a graphing utility to confirm that your sketch is correct. $$y=-2(x+1)^{2}-3$$

Short Answer

Expert verified
The graph is a downward-opening parabola with vertex (-1,-3), vertically compressed by 2.

Step by step solution

01

Identify Base Graph

The base graph is the function \( y = x^2 \), which is a parabola opening upwards with its vertex at the origin (0,0).
02

Horizontal Translation

The term \((x+1)^2\) indicates a horizontal shift. Since \(x\) is replaced with \(x+1\), the graph shifts to the left by 1 unit. The new vertex becomes (-1,0).
03

Vertical Compression and Reflection

The coefficient \(-2\) in \(-2(x + 1)^2\) both compresses the graph vertically and reflects it across the x-axis. The parabola is now upside down (opens downward) and twice as steep. The vertex remains at (-1,0) but the graph is narrower and inverted.
04

Vertical Translation

Finally, the \(-3\) at the end of the equation \(-2(x+1)^2 - 3\) shifts the entire graph down by 3 units. The vertex is now at (-1,-3).
05

Sketch the Graph

Using the transformations identified, sketch the parabola on a coordinate plane. Start at the new vertex (-1,-3), plot additional points if necessary, and ensure it opens downward and is narrower than \( y = x^2\).
06

Confirm with Graphing Utility

Use a graphing utility to plot the equation \( y = -2(x+1)^2 - 3 \) to confirm the accuracy of your sketch. The graph on the utility should match your drawing.

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
A parabola is a U-shaped curve that is the graph of a quadratic function. The standard form of a parabola is given by the equation \(y = ax^2+bx+c\). In its simplest form, such as \(y=x^2\), the parabola is symmetrical and opens upwards with its vertex located at the origin, (0,0).
  • **Vertex:** The highest or lowest point of the parabola, depending on its orientation.
  • **Axis of Symmetry:** A vertical line through the vertex, dividing the parabola into two mirror-image halves.
  • **Direction of Opening:** Determined by the sign of \(a\). If \(a > 0\), it opens upwards; if \(a < 0\), it opens downwards.
The given equation transformation begins with the basic parabola \(y=x^2\). Each subsequent operation affects its shape and position on a coordinate graph. Always keep track of the vertex's movement, as it serves as a reference for all transformations.
Horizontal Translation
Horizontal translation involves shifting the graph left or right. In the equation \((x+1)^2\), \(x\) is replaced by \(x+1\), indicating a shift.
When the form involves \((x+k)^2\), where \(k\) is a constant, the graph moves left by \(k\) units if \(k > 0\) and right if \(k < 0\).
For our problem:
  • Since we have \((x+1)^2\), \(k = 1\), so the graph shifts left by 1 unit.
  • The new vertex from the origin becomes (-1,0).
This step repositions the entire graph without altering its overall shape or steepness, just where it sits horizontally on the coordinate plane.
Vertical Compression
Vertical compression changes the steepness of the parabola. It is impacted by multiplying the quadratic term by a coefficient. In the function \(-2(x + 1)^2\), the coefficient is \(-2\).
This coefficient does double duty:
  • **Compression:** The factor of 2 narrows the parabola, making it steeper than the standard \(y=x^2\).
  • **Reflection:** The negative sign reflects the parabola downwards.
Thus, the graph is not only narrower but inverted. Starting with a vertex at (-1,0), the curve now faces downward and is more pronounced. This modification is crucial in defining how tight or spread out the arch of the parabola appears.
Vertical Translation
Vertical translation involves moving the entire graph up or down along the y-axis. It is dictated by adding or subtracting a constant to the equation. In \(-2(x+1)^2-3\), the term \(-3\) is responsible for this shift.
Here's how it works:
  • **Translation Down:** The subtraction of 3 moves the entire graph down by 3 units.
  • The vertex, previously at (-1,0), is now repositioned to (-1,-3).
This transformation adds another layer of understanding regarding how vertical changes occur independently of other transformations. Together with previous operations, it finalizes the graph's positioning on the coordinate plane. The result is a complete depiction of the parabola's new alignment.

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

In each part, classify the lines as parallel, perpendicular, or neither. (a) \(y=4 x-7\) and \(y=4 x+9\) (b) \(y=2 x-3\) and \(y=7-\frac{1}{2} x\) (c) \(5 x-3 y+6=0\) and \(10 x-6 y+7=0\) (d) \(A x+B y+C=0\) and \(B x-A y+D=0\) (e) \(y-2=4(x-3)\) and \(y-7=\frac{1}{4}(x-3)\)

Suppose that you begin cutting wedge-shaped pieces from a pie so that the are length along the outer crust of each piece is equal to the radius. What fraction of the pie will remain after all pieces that can be cut in this way are eaten?

(a) Use the graph of \(y=\sqrt{x}\) to help sketch the graph of \(y=\sqrt{|x|}\) (b) Use the graph of \(y=\sqrt[3]{x}\) to help sketch the graph of \(y=\sqrt[3]{|x|}\)

Parametric curves can be defined piecewise by using different formulas for different values of the parameter. Sketch the curve that is represented piecewise by the parametric equations $$\begin{aligned}&x=2 t, \quad y=4 t^{2} \quad\left(0 \leq t \leq \frac{1}{2}\right)\\\ &x=2-2 t, \quad y=2 t \quad\left(\frac{1}{2} \leq t \leq 1\right)\end{aligned}$$

Find the slope-intercept form of the equation of the line satisfying the stated conditions, and check your answer using a graphing utility. The line passes through (2,4) and (1,-7).
See all solutions

Recommended explanations on Math Textbooks

Probability and Statistics

Read Explanation

Geometry

Read Explanation

Logic and Functions

Read Explanation

Statistics

Read Explanation

Theoretical and Mathematical Physics

Read Explanation

Calculus

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.