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Chapter 5: Problem 4

Graph \(y=2(x+1)^{2}-3\) using transformations.

Short Answer

Expert verified
Graph shifts left 1 unit, stretches by 2, and shifts down 3 units.

Step by step solution

01

Identify the Parent Function

The given function is a transformation of the parent function for a parabola, which is \(y = x^2\).
02

Apply Horizontal Shift

The equation has \(x + 1\) inside the square term indicating a horizontal shift. The function shifts to the left by 1 unit.
03

Apply Vertical Stretch

The coefficient 2 in front of \( (x + 1)^2 \) indicates a vertical stretch by a factor of 2. This means that the graph will be narrower compared to the parent function.
04

Apply Vertical Shift

The term -3 at the end of the equation indicates a vertical shift down by 3 units.
05

Sketch the Graph

Start with the point \((-1, -3)\) as this is the vertex of the transformed function. Then, apply the stretches and shifts identified in the previous steps to sketch the graph.

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Key Concepts

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

Horizontal Shift
A horizontal shift occurs when the graph of a function moves left or right along the x-axis. This can be easily identified by examining the inside of the function's squared term. In our exercise, we see the term \( x + 1 \).
Vertical Stretch
A vertical stretch or compression modifies how
Vertical Shift
Vertical shifts involve moving the graph up or down along the y-axis. This is shown by a constant added to or subtracted from the function. In our example, the -3 at the end of \( y = 2(x + 1)^2 - 3\)
Parent Function of a Parabola
The parent function of a parabola is the simplest form of a quadratic equation, \( y = x^2 \). It provides the basic shape of a parabola,

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Most popular questions from this chapter

List the potential rational zeros of each polynomial function. Do not attempt to find the zeros. $$ f(x)=3 x^{5}-x^{2}+2 x+18 $$

Solve: \(5 x^{2}-3=2 x^{2}+11 x+1\)

Use the Intermediate Value Theorem to show that each polynomial function has a real zero in the given interval. $$ f(x)=8 x^{4}-2 x^{2}+5 x-1 ;[0,1] $$

Find bounds on the real zeros of each polynomial function. $$ f(x)=-4 x^{5}+5 x^{3}+9 x^{2}+3 x-12 $$

Use the Intermediate Value Theorem to show that the functions \(y=x^{3}\) and \(y=1-x^{2}\) intersect somewhere between \(x=0\) and \(x=1\).
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