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Chapter 1: Problem 78

How will the graph of \(y=-(x-4)^{2}+8\) differ from the graph of \(y=-x^{2} ?\) Check by graphing both functions together.

Short Answer

Expert verified
The graph of \(y=-(x-4)^2 + 8\) is shifted right 4 units and up 8 units, compared to \( y = -x^2 \).

Step by step solution

01

Analyze the Parent Function

The parent function given is \( y = -x^2 \). This is a downward-opening parabola with its vertex at the origin \((0, 0)\).
02

Identify Transformations in the New Function

The function \( y = -(x-4)^2 + 8 \) represents a transformation of the parent function. The expression \((x-4)\) indicates a horizontal shift to the right by 4 units, and the \(+8\) indicates a vertical shift upwards by 8 units.
03

Determine the Vertex of the New Function

The vertex of the parent function \( y = -x^2 \) is \((0, 0)\). After applying the transformations, the vertex of the function \( y = -(x-4)^2 + 8 \) moves to \((4, 8)\).
04

Compare Both Graphs

Both functions are parabolas that open downwards, but the graph of \( y = -(x-4)^2 + 8 \) is shifted 4 units to the right and 8 units up compared to the graph of \( y = -x^2 \).
05

Confirm with Graphing

Graphing both functions confirms the transformations: \( y = -x^2 \) has its vertex at the origin, while \( y = -(x-4)^2 + 8 \) has its vertex at \((4, 8)\) and maintains the same downward-opening shape.

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Key Concepts

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

Quadratic Functions
Quadratic functions are a fundamental concept in algebra and are often expressed in the form \( y = ax^2 + bx + c \). The general shape of a quadratic function鈥檚 graph is a parabola. The orientation and position of this parabola can vary, but it generally has a distinctive U-shape. The key components include:
  • The a value, which affects the parabola's width and orientation (upwards if positive, downwards if negative).
  • The b and c values which influence the parabola's position along the x-axis and y-axis, respectively.
The simplest form of a quadratic function is the standard form \( y = ax^2 \). In our case, the parent function is \( y = -x^2 \). This represents a parabola that opens downward, with its vertex at the origin \((0, 0)\). Understanding the basic shapes and transformations of quadratic functions is crucial for graphing them effectively and recognizing their shifts or changes.
Vertex Form
The vertex form of a quadratic function is particularly useful for graphing and understanding transformations. A quadratic function in vertex form is written as \( y = a(x-h)^2 + k \). This form highlights:
  • The vertex \((h, k)\), which is the highest or lowest point on the graph.
  • The a value, which determines the direction and width of the parabola (same as in the standard form).
Using vertex form makes it easier to visually identify the transformations directly. In the exercise, the function \( y = -(x-4)^2 + 8 \) is in vertex form. Here, the vertex \((4, 8)\) tells us that the parabola has been shifted 4 units to the right and 8 units up from the parent vertex \((0, 0)\). This movement translates the graph without affecting the basic U-shape of the parabola.
Graphing Transformations
Transformations help us understand how the graph of a quadratic function changes or shifts. These transformations include horizontal and vertical shifts, which are essential for comparing graphs of similar functions. The common transformations include:
  • Horizontal Shifts: Occur when we add or subtract inside the parentheses \((x-h)\). For example, \( (x - 4) \) indicates a shift to the right by 4 units.
  • Vertical Shifts: Occur when adding or subtracting outside the parentheses, such as the \(+ 8\), indicating a shift upwards by 8 units.
  • Reflection: Happens if the coefficient of \(x^2\) is negative, flipping the parabola downwards. Our function \( y = -(x-4)^2 + 8 \) retains this reflection from the parent function \( y = -x^2 \).
Recognizing these transformations is key to graphing intricate functions simply. By identifying shifts and reflections, students can effectively visualize how transformations alter the graph's position on the coordinate plane.

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