Problem 335 (a) rewrite each function in \(f... [FREE SOLUTION]

Chapter 9: Problem 335

(a) rewrite each function in $f(x)=a(x-h)^{2}+k$ form and (b) graph it by using transformations. $$f(x)=-x^{2}+8 x-16$$

Short Answer

Expert verified

Rewritten: $ f(x) = -(x - 4)^2 $, Vertex: (4, 0)

Step by step solution

Identify coefficients

In the given quadratic function, identify the coefficients: \[ f(x) = -x^2 + 8x - 16 \] Here, the coefficients are: $ a = -1 $, $ b = 8 $, and $ c = -16 $.

Complete the square

To rewrite the function in vertex form $ f(x) = a(x-h)^2 + k $, start by completing the square. First, factor out $ -1 $ from the first two terms: \[ f(x) = - (x^2 - 8x) - 16 \]Next, add and subtract the square of half of the coefficient of $ x $ inside the parentheses: \[ f(x) = - (x^2 - 8x + 16 - 16) - 16 \] Simplify by combining the constants outside the parentheses:\[ f(x) = - (x - 4)^2 + 16 - 16 \] \[ f(x) = - (x - 4)^2 \]

Identify the vertex

From the vertex form $ f(x) = a(x-h)^2 + k $, we can identify the vertex. Here, $ a = -1 $, $ h = 4 $, and $ k = 0 $. Therefore, the vertex is $ (4, 0) $.

Graph the function

To graph $ f(x) = - (x - 4)^2 $, follow these steps: 1. Start with the parent function $ y = -x^2 $, which is a parabola opening downward. 2. Shift the parabola 4 units to the right to get the vertex at $ (4, 0) $. The vertex form makes it easy to apply these transformations directly.

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Key Concepts

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

Completing the Square

Completing the square is a method used to transform a quadratic function into a form that makes it easier to identify key features of its graph, like the vertex. Let's break down the steps to complete the square:

Start with your quadratic expression: $ ax^2 + bx + c $. In our example, it's $ -x^2 + 8x - 16 $.
Factor out the coefficient of $ x^2 $ from the first two terms. Here, we factor out $ -1 $: $ - (x^2 - 8x) - 16 $.
Next, focus on the quadratic inside the parentheses: $ x^2 - 8x $. To complete the square, add and subtract $ (\frac{b}{2})^2 $ inside the parentheses: $ x^2 - 8x + 16 - 16 $.
Rewrite the quadratic as a perfect square trinomial and simplify: $ - (x - 4)^2 $.

You can now see that through completing the square, the function is transformed and easier to work with.

Vertex Form

The vertex form of a quadratic function is $ f(x) = a(x-h)^2 + k $. This form is particularly useful because it reveals the vertex of the parabola directly. Here are the steps to convert to vertex form:

First, complete the square on the quadratic expression, as detailed in the previous section. For our function, this gave us $ - (x - 4)^2 $.
Once in the form $ a(x-h)^2 + k $, you can identify the vertex $ (h, k) $. In this example, $ h = 4 $ and $ k = 0 $, making the vertex $ (4, 0) $.

The vertex form not only tells you the vertex but also provides information about the direction and width of the parabola. Here, $ a = -1 $ implies the parabola opens downwards.

Graph Transformations

Graph transformations allow you to take the simple $ y = x^2 $ parabola and shift, stretch, or compress it to fit any quadratic function. Here鈥檚 how transformations were applied to our function:

Start with the parent function $ y = -x^2 $, noting it opens downward because of the negative sign.
The transformation $ (x - 4) $ inside the squared term moves the graph 4 units to the right.
There鈥檚 no $k $ term, so there鈥檚 no vertical shift in this example.

These steps help you visualize how the algebraic manipulation affects the graph directly.

Parabolas

A parabola is the graph of a quadratic function, and it has distinct features based on its equation's form. Here's what to look for:

The vertex $ (h, k) $: The turning point of the parabola. In our case, the vertex is $ (4, 0) $.
Direction: Determined by the coefficient $ a $. Here, $ a = -1 $ means the parabola opens downward.
Axis of Symmetry: The vertical line passing through the vertex $ x = h $. For our function, $ x = 4 $.

Understanding these features helps you quickly graph and interpret quadratics. Parabolas occur not only in math but in real-world scenarios like projectile motion and satellite dishes.

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91影视

(a) rewrite each function in \(f(x)=a(x-h)^{2}+k\) form and (b) graph it by using transformations. $$f(x)=-x^{2}+8 x-16$$

Short Answer

Step by step solution

Identify coefficients

Complete the square

Identify the vertex

Graph the function

Key Concepts

Completing the Square

Vertex Form

Graph Transformations

Parabolas

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