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Chapter 2: Problem 11

Express \(f(x)\) in the form \(a(x-h)^{2}+k\) $$f(x)=-\frac{3}{4} x^{2}+9 x-34$$

Short Answer

Expert verified
The vertex form of \(f(x)\) is \(-\frac{3}{4}(x - 2)^2 - 31\).

Step by step solution

01

Identify the Quadratic Function Terms

The given quadratic function is \(f(x) = -\frac{3}{4} x^{2} + 9x - 34\). This can be related to the standard quadratic form \(ax^2 + bx + c\), where \(a = -\frac{3}{4}\), \(b = 9\), and \(c = -34\).
02

Complete the Square

To express \(f(x)\) in vertex form \(a(x-h)^2 + k\), we need to complete the square. First, factor \(-\frac{3}{4}\) out of the first two terms.\[-\frac{3}{4}(x^2 - \frac{12}{3}x) - 34\]Next, complete the square by finding the term needed to complete the perfect square trinomial. Take half of \(-\frac{12}{3}\), square it, and add and subtract it inside the parentheses:\[-\frac{3}{4}\left((x - \frac{12}{3\cdot2})^2 - (\frac{12}{3\cdot2})^2\right) - 34\]\[-\frac{3}{4}\left((x - 2)^2 - 4\right) - 34\]
03

Simplify the Expression

Distribute \(-\frac{3}{4}\) across the terms inside the parentheses and then adjust the constant term:\[-\frac{3}{4}(x - 2)^2 + \frac{3}{4} \times 4 - 34\]\[-\frac{3}{4}(x - 2)^2 + 3 - 34\]This simplifies to:\[-\frac{3}{4}(x - 2)^2 - 31\]
04

Write the Final Vertex Form

Thus, the expression \(f(x)\) in vertex form is:\(f(x) = -\frac{3}{4}(x - 2)^2 - 31\).Where \(a = -\frac{3}{4}\), \(h = 2\), and \(k = -31\).

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Key Concepts

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

Quadratic Function
A quadratic function is a type of polynomial equation that can be expressed in the standard form, \(ax^2 + bx + c\), where \(a\), \(b\), and \(c\) are constants and \(a eq 0\). This equation represents a parabola on the coordinate plane. Here are a few key characteristics of a quadratic function:
  • It has a degree of 2, which means the highest power of \(x\) is 2.
  • The graph of a quadratic function is a curve called a parabola.
  • The parabola can open upwards or downwards depending on the sign of \(a\). If \(a > 0\), it opens upwards, and if \(a < 0\), it opens downwards.
  • The vertex of the parabola is a significant point that represents the maximum or minimum value of the function.
Quadratic functions are used in various real-world contexts, such as projectile motion, where they help to model the path of objects.
Completing the Square
Completing the square is a useful algebraic method for rewriting a quadratic function to reveal important features like its vertex. This technique transforms a quadratic from standard form \(ax^2 + bx + c\) into vertex form \(a(x-h)^2+k\). Let鈥檚 break down the steps involved:
  • First, if \(a\) is not 1, factor \(a\) out from the first two terms. In our example, this means \(-\frac{3}{4}\) is factored out.
  • Next, take the coefficient of \(x\) (after factoring, if necessary), divide it by 2, and square it. This value is crucial for forming a perfect square trinomial.
  • Add and subtract this squared value inside the parentheses to complete the square. Remember, you should balance the equation by distributing and adjusting terms outside the trinomial as needed.
  • The expression inside the parentheses can now be written as \((x-h)^2\), where \(h\) is the value you added and subtracted.
Completing the square is especially valuable because it makes the vertex of the function easy to identify.
Standard Quadratic Form
The standard quadratic form is the common way of presenting quadratic functions: \(ax^2 + bx + c\). Each part of the formula plays a distinct role:
  • \(a\): Determines the direction and "width" of the parabola. A larger absolute value of \(a\) makes the parabola narrower, while a smaller absolute value makes it wider.
  • \(b\): Affects the location of the axis of symmetry and, paired with \(a\), impacts the vertex's horizontal position.
  • \(c\): Represents the y-intercept, the point where the graph intersects the y-axis.
This form is particularly useful for quickly determining the parabola's initial position and direction. However, it doesn't explicitly show the vertex. For this, converting to vertex form using methods like completing the square is necessary. The transformation from standard to vertex form lays the foundation to better analyze parabolas.

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Most popular questions from this chapter

Find a composite function form for \(y\). $$y=\sqrt[4]{x^{4}-16}$$

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