/*! 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 \(g(x)=-2(x+1)^2-2\)... [FREE SOLUTION] | 91影视

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Chapter 2: Problem 39

\(g(x)=-2(x+1)^2-2\)

Short Answer

Expert verified
The constants for the function are \(a=-2\), \(h=-1\), and \(k=-2\). The vertex of the parabola is at the point \((-1,-2)\) and the parabola opens downwards.

Step by step solution

01

STEP 1: Identify the Constants

In the given function \(g(x)=-2(x+1)^2-2\), the constants are as follows: \(a=-2\) (which will flip the parabola vertically and stretch it by a factor of 2), \(h=-1\) (which will shift the parabola 1 unit to the left), and \(k=-2\) (which will shift the parabola 2 units down).
02

STEP 2: Identify the Vertex of the Parabola

The vertex of a parabola in the form \(g(x) = a(x-h)^2 + k\) is at the point \((h,k)\). So for the function \(g(x)=-2(x+1)^2-2\), the vertex of the parabola is at the point \((-1,-2)\).
03

Determine Whether Parabola Opens Upwards or Downwards

Since the value of \(a\), which is -2, is less than zero, the parabola will open downwards.
04

Sketch the Parabola

Plot the vertex at \((-1,-2)\). Since the parabola opens downwards and is stretched by a factor of 2, points on the parabola will be twice as far from the vertex as they are in the basic \(y=x^2\) parabola. Hand draw or use a graphing tool to sketch the function.

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Key Concepts

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

Parabola Vertex Form
Understanding the vertex form of a quadratic function is crucial for easily identifying key features, such as the vertex. The vertex form of a quadratic function is given by:
  • \(g(x) = a(x-h)^2 + k\)
Here:
  • \(a\) determines the direction and the shape of the parabola,
  • \((h, k)\) is the vertex of the parabola.
For the function \(g(x) = -2(x+1)^2-2\), you can directly see that the transformation is in vertex form. The vertex \((h, k)\) in this function is \((-1, -2)\), indicating the parabola's highest or lowest point. Knowing this form helps in quickly visualizing how the parabola is situated on the graph.
Transformations of Quadratic Functions
Transformations can move a quadratic function around the coordinate plane. They include shifting, reflecting, and stretching or compressing. Transformation of a quadratic function is based on the values of \(h\), \(k\), and \(a\) in the vertex form \(g(x) = a(x-h)^2 + k\).

In our function \(g(x)=-2(x+1)^2-2\):
  • The term \(x+1\) indicates a horizontal shift to the left by 1 unit, because \(h = -1\).
  • The \(-2\) outside the squared term shifts the graph down by 2 units as \(k = -2\).
After applying these shifts, the entire graph is relocated on the plane, effectively placing the vertex at the point \((-1, -2)\). These movements help in obtaining the correct position of the parabola on the graph.
Reflection and Stretching of Parabolas
When working with quadratics, the value of \(a\) in \(g(x)=a(x-h)^2+k\) plays a vital role in how the parabola looks. In the equation \(g(x) = -2(x+1)^2-2\), \(a = -2\) does two main things:
  • It reflects the parabola. Since \(a\) is negative, the parabola opens downwards, making it an inverted "U".
  • The magnitude of \(a\), which is 2 here, indicates vertical stretching. The parabola will be narrower on the graph compared to \(y = x^2\), as points move further away from the vertex at a faster rate, stretched by this factor of 2.
Both reflection and stretching are essential to predict how the graph appears without plotting every point manually. This understanding gives a visual guide of the parabola's behavior.

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Most popular questions from this chapter

\(y=4(x-7)^2\)

THOUGHT PROVOKING A jump on a pogo stick with a conventional spring can be modeled by f(x) = ?0.5(x ? 6)2 + 18, where x is the horizontal distance (in inches) and f(x) is the vertical distance (in inches). Write at least one transformation of the function and provide a possible reason for your transformation.

\(y=\frac{1}{2}(x+1)^2+3\)

\(f(x)=\frac{1}{2}(x-2)(x+6)\)

ABSTRACT REASONING A function is written in intercept form with a > 0\. What happens to the vertex of the graph as a increases? as a approaches 0?
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