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Chapter 9: Problem 328

Rewrite each function in the \(f(x)=a(x-h)^{2}+k\) form by completing the square. $$f(x)=-4 x^{2}-16 x-9$$

Short Answer

Expert verified
The function in vertex form is \(f(x) = -4(x+2)^{2} + 7\).

Step by step solution

01

Factor out the leading coefficient

The given function is \(f(x)=-4x^{2}-16x-9\). First, factor out the coefficient of \(x^{2}\) from the terms involving \(x\): \(f(x)=-4(x^{2}+4x)-9\).
02

Complete the square

To complete the square inside the parenthesis, take the coefficient of \(x\), which is 4, divide by 2, and then square it: \((4/2)^2 = 4\). Add and subtract this square inside the parenthesis: \(f(x) = -4(x^2 + 4x + 4 - 4) - 9\).
03

Simplify inside the parentheses

Rewrite the expression by combining the perfect square trinomial: \(f(x) = -4((x+2)^2 - 4) - 9\).
04

Distribute the factored term

Expand the expression by distributing \(-4\) through the terms in the parentheses: \(f(x) = -4(x+2)^{2} + 16 - 9\).
05

Combine constants

Finally, combine the constants to get the function in the required form \(f(x) = -4(x+2)^2 + 7\).

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Key Concepts

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

quadratic functions
A quadratic function is a second-degree polynomial function of the form \( f(x) = ax^2 + bx + c \). Quadratic functions have a characteristic curve called a parabola. These functions are very important in various fields of science, engineering, and mathematics.

Key attributes of quadratic functions include:
  • Standard Form: The standard form is written as \( f(x) = ax^2 + bx + c \). Here, \( a \), \( b \), and \( c \) are constants.
  • Vertex Form: The vertex form is written as \( f(x) = a(x-h)^2 + k \). This form highlights the vertex of the parabola, which is the point \( (h, k) \).
  • Factored Form: The factored form is written as \( f(x) = a(x - r1)(x - r2) \), where \( r1 \) and \( r2 \) are the roots of the polynomial.
Understanding these forms helps in graphing the function and solving various problems involving quadratic equations.
vertex form
The vertex form of a quadratic function provides a convenient way to understand the parabola's properties, especially its vertex. The vertex form of a quadratic function is given by:

\[ f(x) = a(x-h)^2 + k \]

Here, \( (h, k) \) represents the vertex of the parabola, and \( a \) determines the direction and the width of the parabola. Here's how each component affects the parabola:
  • \( h \): Moves the parabola left or right. If \( h \) is positive, the shift is to the right. If \( h \) is negative, the shift is to the left.
  • \( k \): Moves the parabola up or down. If \( k \) is positive, the shift is upward. If \( k \) is negative, the shift is downward.
  • \( a \): Determines the direction and the width of the parabola. If \( a \) is positive, the parabola opens upwards. If \( a \)is negative, it opens downwards. The larger the absolute value of \( a \), the narrower the parabola.
This form is especially useful for graphing and understanding the transformations of quadratic functions.
factoring
Factoring is a crucial algebraic skill that involves writing a polynomial as a product of its factors. This is particularly useful for solving quadratic equations. For a quadratic function in standard form \( f(x) = ax^2 + bx + c \), factoring aims to express it as:

\[ f(x) = a(x - r1)(x - r2) \]

Here, \( r1 \) and \( r2 \) are the roots of the polynomial, the values of \( x \) that make the expression zero:
  • Find the roots: Use methods like the quadratic formula \( x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \) to determine the roots.
  • Verify the factors: Substitute the roots back into the polynomial to ensure they work.
  • Rewrite the function: Once the roots are identified, rewrite the function as a product of linear factors and any common factor \( a \).
Factoring not only helps in solving quadratic equations but also in simplifying expressions and finding the roots of polynomials. However, not all quadratics can be easily factored, leading to methods like completing the square or using the quadratic formula.

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Most popular questions from this chapter

Describe the steps needed to solve a quadratic inequality graphically.

Solve each inequality algebraically and write any solution in interval notation. $$x^{2}+x-6 \leq 0$$

Solve each inequality algebraically and write any solution in interval notation. $$-x^{2}+2 x-7 \geq 0$$

(a) rewrite each function in \(f(x)=a(x-h)^{2}+k\) form and (b) graph it by using transformations. $$f(x)=3 x^{2}-6 x-1$$

Rewrite each function in the \(f(x)=a(x-h)^{2}+k\) form by completing the square. $$f(x)=2 x^{2}-12 x+7$$
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