Problem 11 In exercises $5-11,$ write eac... [FREE SOLUTION]

Chapter 2: Problem 11

In exercises $5-11,$ write each function in the form $(x+a)^{2}+b$ and identify the values of $a$ and $b$. $$ p(x)=x^{2}+13 x $$

Short Answer

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Step by step solution

Completing the Square

To convert the function into the form $(x + a)^2 + b$, we need to complete the square. Start by focusing on the quadratic $(x^2 + 13x)$. Half of the linear coefficient (13) is $(13/2)$, so you square this to complete the square: $\left(\frac{13}{2}\right)^2 = \frac{169}{4}.$

Rewriting the Function

Add and subtract $(\frac{169}{4})$ inside the expression to maintain the equality: $ p(x) = x^2 + 13x + \frac{169}{4} - \frac{169}{4}.$Factor the perfect square trinomial $(x^2 + 13x + \frac{169}{4})$ into $ \left(x + \frac{13}{2}\right)^2 $. Now rewrite the function: $ p(x) = \left(x + \frac{13}{2}\right)^2 - \frac{169}{4}.$

Identifying a and b

The function is now in the form $ (x + a)^2 + b $. By comparing $ \left(x + \frac{13}{2}\right)^2 - \frac{169}{4} $with $ (x+a)^2 + b $, you identify $ a = \frac{13}{2} $ and $ b = -\frac{169}{4} $.

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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 are defined by the general form $ ax^2 + bx + c $, where $ a $, $ b $, and $ c $ are constants, and $ a eq 0 $. These functions graph into a shape known as a parabola. These parabolas can open either upwards or downwards, depending on the sign of the coefficient $ a $.

Quadratic functions are incredibly useful for modeling various real-world scenarios, such as projectile motion and the economics of profit and cost relationships. Understanding their properties, like the vertex, axis of symmetry, and direction of opening, is essential for analyzing their behavior. By converting quadratics into different forms, such as vertex form, the nature and solutions of these functions become clearer.

Factoring

Factoring involves breaking down a quadratic expression into simpler, multiplicative components. For instance, a quadratic like $ x^2 + 13x $ is often factorable into expressions of two binomials. However, since $ x^2 + 13x $ does not initially appear as a factorable expression, we rely on the method of completing the square to transform it.

The factored form is particularly useful for identifying the roots of the quadratic function, which are the x-values that make the function zero. Using methods like the quadratic formula or finding the square will help confirm these factors, ensuring accuracy and understanding.

Perfect Square Trinomial

A perfect square trinomial arises when a trinomial can be expressed as the square of a binomial. In simpler words, it can be rearranged into the form $(x + d)^2$. This is achieved through completing the square.

To complete the square, start with half of the linear coefficient, square it, and adjust the original expression by adding and subtracting this square. For example, with $ x^2 + 13x $, the linear coefficient is 13. Half of this, squared $ (\frac{13}{2})^2 $, gives $ \frac{169}{4} $. This transition helps convert the quadratic into the perfect square trinomial form. This makes it easier to understand and graphically represent the quadratic function.

Converting Functions

Converting functions into different forms, particularly into vertex form $(x + a)^2 + b$, helps simplify the analysis of a quadratic function's behavior. It is achieved through the method of completing the square, which reveals the vertex of a parabola, simplifying the graphing and interpreting process.

For example, converting $ p(x) = x^2 + 13x $ involves completing the square to express it as $ (x + \frac{13}{2})^2 - \frac{169}{4} $. Here, the values $ a = \frac{13}{2} $ and $ b = -\frac{169}{4} $ indicate the shift from the origin, making it easier to draw and understand the function's graph. Converting functions in this manner aids significantly in solving equations and interpreting the solutions.

91影视

In exercises \(5-11,\) write each function in the form \((x+a)^{2}+b\) and identify the values of \(a\) and \(b\). $$ p(x)=x^{2}+13 x $$

Short Answer

Step by step solution

Completing the Square

Rewriting the Function

Identifying a and b

Key Concepts

Quadratic Functions

Factoring

Perfect Square Trinomial

Converting Functions

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