/*! This file is auto-generated */ .wp-block-button__link{color:#fff;background-color:#32373c;border-radius:9999px;box-shadow:none;text-decoration:none;padding:calc(.667em + 2px) calc(1.333em + 2px);font-size:1.125em}.wp-block-file__button{background:#32373c;color:#fff;text-decoration:none} Problem 12 Find the vertex of each parabola... [FREE SOLUTION] | 91影视

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Chapter 9: Problem 12

Find the vertex of each parabola. \(f(x)=x^{2}-x+5\)

Short Answer

Expert verified
(\frac{1}{2}, \frac{19}{4})

Step by step solution

01

Identify the coefficients

For the quadratic function in standard form, identify the coefficients. In the expression \( f(x) = ax^2 + bx + c \), the coefficients are:\(a = 1\), \(b = -1\), and \(c = 5\).
02

Use the vertex formula

The x-coordinate of the vertex for a quadratic function is found using the formula: \( x = \frac{-b}{2a} \). Plug in the identified coefficients:\( x = \frac{-(-1)}{2(1)} \).
03

Simplify the x-coordinate

Simplify the expression found in Step 2:\( x = \frac{1}{2} \).
04

Find the y-coordinate

Substitute the x-coordinate back into the original function to find the y-coordinate:\( f(\frac{1}{2}) = (\frac{1}{2})^2 - \frac{1}{2} + 5 \).
05

Simplify the y-coordinate

Calculate the result:\( f(\frac{1}{2}) = \frac{1}{4} - \frac{1}{2} + 5 = \frac{1}{4} - \frac{2}{4} + \frac{20}{4} = \frac{19}{4} \).
06

State the vertex

Combine the x- and y-coordinates to express the vertex:The vertex of the parabola is \( (\frac{1}{2}, \frac{19}{4}) \).

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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 function. It is expressed in the general form:
\(f(x) = ax^2 + bx + c\).
Here:
  • \(a\) is the coefficient of the quadratic term \(x^2\)
  • \(b\) is the coefficient of the linear term \(x\)
  • \(c\) is the constant term
This type of function forms a parabola when graphed.

A key property of a quadratic function is that its highest or lowest point (depending on the direction it opens) is the vertex.
The parabolas open upwards if \(a > 0\) and downwards if \(a < 0\).
The axis of symmetry divides the parabola into two symmetric halves. It runs vertically through the vertex.

Understanding these properties helps to identify the vertex and the graph of the function.
vertex formula
The vertex formula allows us to find the vertex of a parabola.
For a quadratic function given by \(f(x) = ax^2 + bx + c\), the vertex formula is used to determine the x-coordinate of the vertex.

The formula is:
\[ x = \frac{-b}{2a} \]
Here:
  • \

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Most popular questions from this chapter

Compared to the graph of \(f(x)=\sqrt{x}\), the graph of \(g(x)=\frac{1}{2} \sqrt{x}\) is (stretched / shrunken) by a factor of ______. Some points on the graph of \(f(x)\) are (0,0),(4,2) , and (16,4) . Corresponding points on the graph of \(g(x)\) are (0,0) (4, ____), and (16, ____).

Suppose that postage rates are \(\$ 0.55\) for the first ounce, plus \(\$ 0.24\) for each additional ounce, and that each letter carries one \(\$ 0.55\) stamp and as many \(\$ 0.24\) stamps as necessary. Graph the function \(y=p(x)=\) the number of stamps on a letter weighing \(x\) ounces. Use the interval (0,5] .

Sketch each graph. $$ g(x)=4 \sqrt{x} $$

In each problem, find the following. (a) A function \(R(x)\) that describes the total revenue received (b) The graph of the function from part (a) (c) The number of unsold seats that will produce the maximum revenue (d) The maximum revenue A charter flight charges a fare of \(\$ 200\) per person, plus \(\$ 4\) per person for each unsold seat on the plane. The plane holds 100 passengers. Let \(x\) represent the number of unsold seats. (Hint: To find \(R(x),\) multiply the number of people flying, \(100-x\), by the price per ticket, \(200+4 x\).)

The tables give some selected ordered pairs for functions \(f\) and \(g\). $$\begin{array}{r|r}{x} & f(x) \\\\\hline-1 & 1 \\\\\hline 2 & -1 \\\\\hline 5 & 9\end{array}$$ $$\begin{array}{c|c}x & g(x) \\\\\hline 2 & -1 \\\\\hline 7 & 5 \\\\\hline 1 & 9 \\\\\hline 9 & 20\end{array}$$ Find each of the following. $$ (g \circ g)(1) $$
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