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Chapter 1: Problem 366

For the following exercises, describe how the formula is a transformation of a toolkit function. Then sketch a graph of the transformation. $$ g(x)=4(x+1)^{2}-5 $$

Short Answer

Expert verified
\( g(x) = 4(x+1)^2 - 5 \) is a transformed \( x^2 \) with shifts and a stretch. The vertex is at \((-1, -5)\), and the parabola is narrower.

Step by step solution

01

Identify the Toolkit Function

The given function is a transformation of the toolkit function \( f(x) = x^2 \), which is the basic quadratic function.
02

Determine the Horizontal Translation

The function is \( g(x) = 4(x+1)^2 - 5 \). Notice the \((x+1)\) in the expression. This indicates a horizontal shift of the graph left by 1 unit.
03

Determine the Vertical Translation

The constant term outside the squared expression is \(-5\). This results in a vertical shift of the graph down by 5 units.
04

Determine the Vertical Stretch

The coefficient in front of the squared term is 4. This means the graph will be vertically stretched by a factor of 4. The "4" makes the parabola narrower compared to the original \( f(x) = x^2 \).
05

Combine the Transformations

Combine all three transformations: shift the graph of \( f(x) = x^2 \) left by 1 unit, down by 5 units, and apply a vertical stretch by a factor of 4. The graph of \( g(x) = 4(x+1)^2 - 5 \) is a narrow parabola that opens upwards with its vertex at \(( -1, -5 )\).
06

Sketch the Graph

Draw the new graph based on the transformations identified. The vertex will move from \((0, 0)\) to \((-1, -5)\) due to the transformations. The graph is narrower due to the stretch.

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Key Concepts

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

Quadratic Function
A quadratic function is one of the simplest types of polynomial functions. It takes the form \( f(x) = ax^2 + bx + c \), where \( a \), \( b \), and \( c \) are constants, and \( a eq 0 \). The graph of a quadratic function is a parabola.
  • If \( a > 0 \), the parabola opens upwards.
  • If \( a < 0 \), the parabola opens downwards.
The simplest quadratic function is \( f(x) = x^2 \), which is known as the "toolkit" function. Its graph is a symmetric parabola that has its vertex at the origin \( (0, 0) \). Understanding this basic function helps in identifying transformations and manipulations that apply to more complex quadratic functions.
Horizontal Translation
Horizontal translation refers to moving the graph of a function left or right along the x-axis. It is accomplished by adding or subtracting a constant from the variable \( x \) inside the function.
In the function \( g(x) = 4(x+1)^2 - 5 \), we observe a horizontal translation. The replacement of \( x \) with \( (x+1) \) indicates that the function is shifted to the left by 1 unit.
  • Adding a positive number: \( (x - h) \), moves the graph to the right by \( h \) units.
  • Adding a negative number: \( (x + h) \), moves the graph to the left by \( h \) units.
This type of transformation does not affect the shape of the graph, only its position.
Vertical Translation
Vertical translation involves shifting the graph of a function up or down along the y-axis. This is achieved by adding or subtracting a constant outside the function. In our function \( g(x) = 4(x+1)^2 - 5 \), the \(-5\) at the end signifies a downward shift.
  • Adding a constant: \( +k \), raises the graph by \( k \) units.
  • Subtracting a constant: \( -k \), lowers the graph by \( k \) units.
In this case, the graph is shifted down by 5 units. While the vertex of a parabola normally lies at its origin, due to this translation, the graph of the quadratic function has been pushed down, changing its vertex from \((0, 0)\) to \((-1, -5)\) after considering all transformations.
Vertical Stretch
Vertical stretch is a type of transformation that alters the "height" or "width" of a graph, making it taller and narrower, or shorter and wider. It is governed by the coefficient of the squared term in a quadratic function.
For \( g(x) = 4(x+1)^2 - 5 \), the factor "4" in front of \((x+1)^2\) is responsible for the vertical stretch. This means every point on the basic parabola \( f(x) = x^2 \) is moved 4 times further from the x-axis.
  • If the coefficient is greater than 1, the parabola becomes narrower.
  • If the coefficient is between 0 and 1, the parabola becomes wider.
  • A negative coefficient would flip the parabola upside down (not the case here as the coefficient is positive).
This transformation significantly affects the appearance of the graph due to the increased steepness brought on by the factor of 4, while the original width of the parabola \( f(x) = x^2 \) becomes more compact.

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Most popular questions from this chapter

For the following exercises, describe how the formula is a transformation of a toolkit function. Then sketch a graph of the transformation. $$ a(x)=\sqrt{-x+4} $$

For the following exercises, describe how the formula is a transformation of a toolkit function. Then sketch a graph of the transformation. $$ h(x)=-2|x-4|+3 $$

For the following exercises, determine the interval(s) on which the function is increasing and decreasing. $$ g(x)=5(x+3)^{2}-2 $$

For the following exercises, use each pair of functions to find \(f(g(x))\) and \(g(f(x)) .\) Simplify your answers. $$ f(x)=\frac{1}{x-4}, g(x)=\frac{2}{x}+4 $$

For the following exercises, use each pair of functions to find \(f(g(0))\) and \(g(f(0))\) $$ f(x)=\frac{1}{x+2}, g(x)=4 x+3 $$
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