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Chapter 5: Problem 60

REVIEW Which of the following most accurately describes the translation of the graph of \(y=(x+5)^{2}-1\) to the graph of \(y=(x-1)^{2}+3 ?\) \(\mathbf{F}\) up 4 and 6 to the right \(\mathbf{G}\) up 4 and 1 to the left \(\mathbf{H}\) down 1 and 1 to the right \(\mathbf{J}\) down 1 and 5 to the left

Short Answer

Expert verified
Option F: up 4 and 6 to the right.

Step by step solution

01

Identify Transformation of First Equation

The equation \( y = (x+5)^2 - 1 \) represents a parabola. It can be written in the vertex form \( y = a(x-h)^2 + k \) where \((h, k)\) is the vertex. Here, \((h, k) = (-5, -1)\). This means the parabola starts at \((-5, -1)\).
02

Identify Transformation of Second Equation

The second equation \( y = (x-1)^2 + 3 \) is also in the vertex form \( y = a(x-h)^2 + k \) where \((h, k)\) is the vertex. Here, \((h, k) = (1, 3)\). This means the parabola starts at \((1, 3)\).
03

Calculate Horizontal Shift

Compare the x-coordinates of the vertices. The initial vertex is \( -5 \), and the final vertex is \( 1 \). The horizontal shift is from \(-5\) to \(1\), a movement of \(6\) units to the right.
04

Calculate Vertical Shift

Compare the y-coordinates of the vertices. The initial y-coordinate is \(-1\) and the final y-coordinate is \(3\). The vertical change is from \(-1\) to \(3\), a shift of \(+4\) units upwards.
05

Determine Correct Answer

Summarize the transformations: Move 6 units to the right and 4 units up. This matches option \(\mathbf{F}\): "up 4 and 6 to the right".

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Key Concepts

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

Vertex Form
The vertex form of a quadratic equation is a powerful tool for graph transformations. It is expressed as \( y = a(x-h)^2 + k \). Here, \( h \) and \( k \) determine the vertex of the parabola, \( (h, k) \).

This form reveals key features of the graph, such as its width, direction, and position. By changing \( h \) and \( k \), you can easily shift the graph's position without altering its shape.
  • The value \( a \) affects the direction and "width" or "concavity" of the parabola.
  • When \( a \) is positive, the parabola opens upwards, and if negative, it opens downwards.
  • The vertex \((h, k)\) indicates where the parabola's highest or lowest point is located.
Understanding the vertex form simplifies the process of analyzing transformations of quadratic equations.
Quadratic Equations
Quadratic equations represent one of the most fundamental concepts in algebra. They are polynomial equations of degree 2 and can generally be written as \( ax^2 + bx + c = 0 \).

These equations describe parabolas when graphed, which are U-shaped curves that can open either upward or downward. The shape and position of the parabola can change drastically with transformations.
  • The "standard form" of a quadratic equation is \( ax^2 + bx + c \).
  • They can be converted into "vertex form" to easily analyze transformations and find the vertex.
  • Parabolas always have a vertex, an axis of symmetry, and can have a "maximum" or "minimum" point depending on the direction.
Recognizing these features is crucial for mastering quadratic equations and graph transformations.
Horizontal Shift
Horizontal shifts move the graph of a function left or right on the x-axis. This is done by changing the \( h \) value in the vertex form \( y = a(x-h)^2 + k \).

If you increase \( h \), the graph moves to the right. Conversely, if \( h \) is decreased, it moves to the left. This simple adjustment affects only the x-coordinate of the vertex.
  • Example: Changing \( y = (x+5)^2 - 1 \) to \( y = (x-1)^2 - 1 \) shifts the parabola from \(-5\) to \(1\) on the x-axis, a 6-unit shift to the right.
  • Always think of the term \((x-h)\). If the expression inside changes from \(x+5\) to \(x+1\), it represents a left shift.
Mastering horizontal shifts can help you control how a graph moves parallel along the x-axis.
Vertical Shift
Vertical shifts occur when the graph of a function is moved up or down along the y-axis. In vertex form, it is controlled by the \( k \) value in \( y = a(x-h)^2 + k \).

By adjusting \( k \), you can elevate or lower the graph without changing its shape or horizontal position.
  • Increasing \( k \) moves the graph up, while decreasing \( k \) moves it down.
  • Example: Changing from \( y = (x-1)^2 - 1 \) to \( y = (x-1)^2 + 3 \) results in a 4-unit shift upwards.
Vertical shifts are one of the most straightforward transformations, maintaining the parabola's concavity and symmetry while relocating it vertically.

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