Problem 332 (a) rewrite each function in \(f... [FREE SOLUTION]

Chapter 9: Problem 332

(a) rewrite each function in $f(x)=a(x-h)^{2}+k$ form and (b) graph it by using transformations. $$f(x)=x^{2}-6 x+8$$

Short Answer

Expert verified

Rewritten function: \[f(x) = (x-3)^2 - 1\].Graph using transformations: Shift right by 3 units and down by 1 unit.

Step by step solution

Identify the Quadratic Expression

The given function is: \[f(x) = x^2 - 6x + 8\]Recognize that this is in the standard quadratic form.

Rewrite as a Perfect Square Trinomial

To rewrite the quadratic function in the form \[f(x) = a(x-h)^2 + k\]complete the square.Start with the expression inside the function: \[x^2 - 6x + 8\]Take the coefficient of x (which is -6), divide it by 2, and square the result: \[\frac{-6}{2} = -3\] \[(-3)^2 = 9\]Now rewrite the quadratic expression by adding and subtracting this square term: \[x^2 - 6x + 9 - 9 + 8\]}

Simplify the Expression

Group the perfect square trinomial and simplify the constants: \[(x^2 - 6x + 9) - 9 + 8\]This can be written as: \[(x - 3)^2 - 1\]Thus, rewritten in vertex form: \[f(x) = (x-3)^2 - 1\]}

Identify Transformations

Compare the vertex form \[f(x) = (x-3)^2 - 1\] with \[f(x) = a(x-h)^2 + k\].The vertex is at \[(h, k) = (3, -1)\].This tells you the transformations relative to \[f(x) = x^2\]:- Shift right by 3 units.- Shift down by 1 unit.

Graph the Function

Start with the graph of \[y = x^2\].Apply the transformations:1. Shift the graph right by 3 units, so the vertex moves from \[(0, 0)\] to \[(3, 0)\].2. Shift the graph down by 1 unit, so the vertex moves from \[(3, 0)\] to \[(3, -1)\].The final graph is a parabola opening upwards with vertex at \[(3, -1)\].

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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 fundamental in algebra and pre-calculus. They are represented by polynomial equations of the form: y = ax^2 + bx + c.Here, 'a', 'b', and 'c' are constants, and the highest power of the variable x is 2, making it a second-degree polynomial. The graph of a quadratic function is a parabola, which can either open upwards if a > 0 or downwards if a < 0.Quadratic functions are essential because they model many real-world situations, like the path of a thrown ball or the area of a space. Understanding their properties, such as finding the vertex, axis of symmetry, and roots, is crucial for solving various mathematical problems.

vertex form

The vertex form of a quadratic function is a way to express the function to easily identify its vertex, which is the highest or lowest point on the graph (parabola). The vertex form is written as:f(x) = a(x - h)^2 + k.In this formula:

'a' determines the direction and width of the parabola.
(h, k) defines the vertex of the parabola.

The vertex form is very useful because it reveals the vertex directly, which helps in graphing and understanding the transformations applied to the parent function y = x^2.

graph transformations

Graph transformations allow you to manipulate and shift the graph of a function without changing its fundamental shape.Transformations can include:

Vertical Shifts: Moving the graph up or down without altering its shape. This affects the 'k' value in the vertex form.
Horizontal Shifts: Moving the graph left or right. This affects the 'h' value in the vertex form.
Reflections: Flipping the graph over the x-axis or y-axis, depending on the sign of 'a'.
Vertical Stretch/Compression: Making the graph narrower or wider, controlled by the absolute value of 'a'.

In the given exercise, the function f(x) = x^2 - 6x + 8 was transformed to f(x) = (x - 3)^2 - 1. This indicates a horizontal shift of 3 units to the right and a vertical shift of 1 unit downwards, moving the vertex to (3, -1).

parabola

A parabola is the U-shaped graph of a quadratic function. It can open upwards or downwards depending on the coefficient 'a' in the quadratic function:

If a > 0, the parabola opens upwards.
If a < 0, the parabola opens downwards.

Key characteristics of a parabola include:

Vertex: The highest or lowest point of the parabola, given by (h, k) in the vertex form.
Axis of Symmetry: A vertical line that divides the parabola into two mirror images, passing through the vertex at x = h.
Focus: A point inside the parabola used to define its shape, found using the formula 1/(4a).
Directrix: A line perpendicular to the axis of symmetry, located equidistant from the vertex as the focus, which helps in constructing the parabola using the definition.

Understanding these properties allows you to graph any quadratic function and analyze the transformations applied to it effectively.

91影视

(a) rewrite each function in \(f(x)=a(x-h)^{2}+k\) form and (b) graph it by using transformations. $$f(x)=x^{2}-6 x+8$$

Short Answer

Step by step solution

Identify the Quadratic Expression

Rewrite as a Perfect Square Trinomial

Simplify the Expression

Identify Transformations

Graph the Function

Key Concepts

quadratic functions

vertex form

graph transformations

parabola

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