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Chapter 1: Problem 67

Transform the function into the form \(f(x)=c(x-h)^{2}+k,\) where \(c, k,\) and \(h\) are constants, by completing the square. Use graph-shifting techniques to graph the function. $$f(x)=2 x^{2}-8 x+3$$

Short Answer

Expert verified
Rewrite as \( f(x) = 2(x-2)^2 - 5 \). Graph with vertex at (2, -5).

Step by step solution

01

Identify the Quadratic Terms

We start with the quadratic function given: \( f(x) = 2x^2 - 8x + 3 \). The goal is to rewrite this function in the form \( f(x) = c(x-h)^2 + k \) by completing the square.
02

Factor Out the Coefficient of x虏

The term \( 2x^2 \) has a coefficient of 2. Factor 2 out of the first two terms: \[ f(x) = 2(x^2 - 4x) + 3 \]
03

Complete the Square

To complete the square, focus on the expression inside the parentheses: \( x^2 - 4x \). Take half of the linear term coefficient (-4), which is -2, and square it, resulting in 4. Add and subtract this square inside the parentheses:\[ f(x) = 2(x^2 - 4x + 4 - 4) + 3 \]This simplifies inside the parentheses to:\[ f(x) = 2((x^2 - 4x + 4) - 4) + 3 \]
04

Write as a Perfect Square and Simplify

The expression \( x^2 - 4x + 4 \) is a perfect square: \( (x-2)^2 \). Substitute this back into the equation:\[ f(x) = 2((x-2)^2 - 4) + 3 \]Distribute the 2 and simplify:\[ f(x) = 2(x-2)^2 - 8 + 3 \]\[ f(x) = 2(x-2)^2 - 5 \]
05

Identify c, h, and k

Now, the function is in the form \( f(x) = c(x-h)^2 + k \), where \( c = 2 \), \( h = 2 \), and \( k = -5 \).
06

Graph the Function

Using graph-shifting techniques based on the transformation: - \( c = 2 \): Vertical stretch by a factor of 2.- \( h = 2 \): Shift right by 2 units.- \( k = -5 \): Shift down by 5 units.Plot the vertex at \( (h, k) = (2, -5) \) and sketch the parabola opening upwards.

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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 function that can be written in the form \( f(x) = ax^2 + bx + c \). It's a polynomial of degree two, where the graph of this function forms a parabola. There are some key features that help in understanding quadratic functions better:
  • Standard Form: The traditional representation is \( f(x) = ax^2 + bx + c \).
  • Vertex: The highest or lowest point of the parabola. The location of the vertex helps in graphing the parabola accurately.
  • Axis of Symmetry: A vertical line that runs through the vertex, typically given by the formula \( x = -\frac{b}{2a} \).
  • Direction: If the coefficient \( a \) is positive, the parabola opens upwards. If \( a \) is negative, it opens downwards.
Quadratic functions are fundamental in algebra and are used to model various real-world situations, such as projectile motion and area calculations.
Vertex Form
Vertex form is an alternative way to express a quadratic function. It makes it easy to identify the vertex and thus graph the parabola efficiently. The vertex form of a quadratic function is given by:\[ f(x) = c(x - h)^2 + k \]Here's why vertex form is particularly useful:
  • Immediate Identification: The values \( h \) and \( k \) represent the vertex of the parabola \( (h, k) \).
  • Simplifies Graphing: Once in vertex form, graphing the function involves a simple translation of the basic parabola \( y = x^2 \).
  • Graph Translation Insights: The parameter \( c \) impacts the width and direction of the parabola, while \( h \) and \( k \) indicate horizontal and vertical shifts.
Converting from standard form to vertex form involves completing the square, a technique that allows us to find the perfect square trinomial inside the quadratic expression.
Graph-shifting Techniques
Graph-shifting techniques are useful when graphing transformed equations, particularly those in vertex form. Understanding these helps visualize how a parabola's position changes on a coordinate plane.
  • Horizontal Shift: The value \( h \) in the vertex form \( (x - h)^2 \) moves the graph left or right. If \( h \) is positive, shift right; if negative, shift left.
  • Vertical Shift: The constant \( k \) dictates how much we move up or down. If \( k \) is positive, shift up; if negative, shift down.
  • Vertical Stretch/Compression: The coefficient \( c \) affects the graph's shape. A larger \( |c| \) causes a vertical stretch, making the parabola narrower, while a smaller \( |c| \) results in a compression, making it wider.
By manipulating these parameters, you can effectively sketch any quadratic function based on its vertex form to predict its position and shape on the grid.

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Most popular questions from this chapter

In Exercises \(51-60\), show that \(f(g(x))=x\) and \(g(f(x))=x\). $$f(x)=\frac{x-2}{3}, \quad g(x)=3 x+2$$

In Exercises \(77-81,\) for the functions \(f(x)=x+2\) and \(g(x)=x^{2}-4,\) find the indicated function and state its domain. Explain the mistake that is made in each problem. $$\frac{g}{f}$$ Solution: $$\begin{aligned}\frac{g(x)}{f(x)} &=\frac{x^{2}-4}{x+2} \\\&=\frac{(x-2)(x+2)}{x+2} \\\&=x-2\end{aligned}$$ Domain: \((-\infty, \infty)\) This is incorrect. What mistake was made?

Graph the functions \(f\) and \(g\) and the line \(y=x\) in the same screen. Do the two functions appear to be inverses of each other? $$f(x)=\sqrt[3]{x+3}-2 ; \quad g(x)=x^{3}+6 x^{2}+12 x+6$$

Refer to the following: In calculus, the difference quotient \(\frac{f(x+h)-f(x)}{h}\) of a function fis used to find the derivative \(f^{\prime}\) of \(f\), by allowing \(h\) to approach zero, \(h \rightarrow 0 .\) The derivative of the inverse function \(\left(f^{-1}\right)^{\prime}\) can be found using the formula $$\left(f^{-1}\right)^{\prime}(x)=\frac{1}{f^{\prime}\left(f^{-1}(x)\right)}$$ provided that the denominator is not 0 and both \(f\) and \(f^{-1}\) are differentiable. For the following one-to-one function, find (a) \(f^{-1},\) (b) \(f^{\prime},\) (c) \(\left(f^{-1}\right)^{\prime},\) and (d) verify the formula above. For (b) and (c), use the difference quotient. $$f(x)=2 x+1$$

Suppose that you want to build a square fenced-in area for your dog. Fencing is purchased in linear feet. a. Write a composite function that determines the area of your dog pen as a function of how many linear feet are purchased. b. If you purchase 100 linear feet, what is the area of your dog pen? c. If you purchase 200 linear feet, what is the area of your dog pen?
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