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Chapter 3: Problem 28

Solve each quadratic inequality by locating the \(x\) -intercept(s) (if they exist), and noting the end behavior of the graph. Begin by writing the inequality in function form as needed. $$-x^{2}+3 x<3$$

Short Answer

Expert verified
The inequality is true for all real x.

Step by step solution

01

Write in Function Form

The first step is to write the inequality \(-x^{2}+3x<3\) in function form. We need to rearrange it to look like a quadratic function by bringing all the terms to one side of the inequality.Subtract 3 from both sides:\[-x^2 + 3x - 3 < 0\]
02

Identify as Quadratic Function

Recognize the left-hand side as a quadratic function, i.e., \(f(x) = -x^2 + 3x - 3\).We'll now study this quadratic to solve the inequality.
03

Solve the Equation for x-intercepts

To find the x-intercepts, solve the equation \(-x^2 + 3x - 3 = 0\).Rearrange into standard quadratic form: \(x^2 - 3x + 3 = 0\).Use the quadratic formula \(x = \frac{-b \pm \sqrt{b^2-4ac}}{2a}\) where \(a = 1\), \(b = -3\), \(c = 3\).\[ x = \frac{-(-3) \pm \sqrt{(-3)^2-4(1)(3)}}{2(1)} \]\[ x = \frac{3 \pm \sqrt{9-12}}{2} \]\[ x = \frac{3 \pm \sqrt{-3}}{2} \]The discriminant is negative, meaning there are no real x-intercepts.
04

Analyze End Behavior

Examine the end behavior of the quadratic function \(f(x) = -x^2 + 3x - 3\).Since the leading coefficient \(-1\) is negative, the parabola opens downwards. As \(x\) approaches infinity or negative infinity, \(f(x)\) approaches negative infinity.
05

Conclusion Based on Analysis

The function \(f(x)\) is a downward-opening parabola without real x-intercepts, which means it's always below the x-axis.Therefore, \(-x^2 + 3x - 3 < 0\) is satisfied for all real values of \(x\).

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Key Concepts

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

x-intercepts
Finding the x-intercepts of a quadratic function means identifying when the output, or f(x), is zero. These intercepts are the points where the parabola crosses the x-axis. In the equation \(-x^{2} + 3x - 3 = 0\), we rearrange it into the standard form \(x^{2} - 3x + 3 = 0\) and try to find solutions using the quadratic formula.

However, if the discriminant (the expression under the square root in the quadratic formula) is negative, there are no real solutions, meaning the function does not cross the x-axis. In our case, the discriminant is \(-3\), confirming no real x-intercepts exist for our function. This tells us that the parabola stays entirely above or below the x-axis.
end behavior
The end behavior of a quadratic function informs us how the function behaves as the input variable, x, moves towards positive or negative infinity. By observing the leading term of the quadratic function, specifically its coefficient and sign, we can determine this behavior.

In the quadratic function \(f(x) = -x^{2} + 3x - 3\), the leading term is \(-x^{2}\). With a negative leading coefficient, the parabola opens downward. This means that both ends of the parabola will approach negative infinity as x moves towards positive or negative infinity.
  • If the leading coefficient is positive, the parabola opens upwards, heading towards positive infinity.
  • If it鈥檚 negative, like in our function, the ends drop down toward negative infinity.
Understanding end behavior helps predict the graph's direction based solely on the quadratic's leading term.
parabola
A parabola is a U-shaped plot that represents a quadratic function visually. It can open upwards or downwards based on the leading coefficient of the quadratic term. The vertex forms the parabola's peak if it opens downward, or its lowest point if it opens upward.

For our quadratic function \(f(x) = -x^{2} + 3x - 3\), the leading coefficient is \(-1\), indicating that the parabola opens downwards. This down opening shape confirms that as x values increase or decrease, the function value f(x) drops below the x-axis.
  • When no real x-intercepts are present, the entire parabola can either sit above or fall below the x-axis, depending on the direction it opens.
Understanding the orientation of a parabola helps in predicting its positioning relative to the x-axis and solving inequalities involving quadratic functions.
quadratic function
A quadratic function is any function in the standard form \(ax^{2} + bx + c\), where a, b, and c are constants, and a is not zero. These functions create parabolic graphs and can model various real-world scenarios due to their distinctive U-shape.

Given the function \(f(x) = -x^{2} + 3x - 3\), we identify it as quadratic because its highest degree term is \(x^{2}\). The leading term here (\

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Most popular questions from this chapter

A large group of students is asked to memorize a list of 50 Italian words, a language that is unfamiliar to them. The group is then tested regularly to see how many of the words are retained over a period of time. The average number of words retained is modeled by the function \(W(t)=\frac{6 t+40}{t},\) where \(W(t)\) represents the number of words remembered after \(t\) days. a. Graph the function over the interval \(t \in[0,40] .\) How many days until only half the words are remembered? How many days until only one-fifth of the words are remembered? b. After 10 days, what is the average number of words retained? How many days until only 8 words can be recalled? c. What is the significance of the horizontal asymptote (what does it mean in this context)?

Tina and Imai have just purchased a purebred German Shepherd, and need to fence in their backyard so the dog can run. What is the maximum rectangular area they can enclose with \(200 \mathrm{ft}\) of fencing, if (a) they use fencing material along all four sides? What are the dimensions of the rectangle? (b) What is the maximum area if they use the house as one of the sides? What are the dimensions of this rectangle?

Wages earned varies directly with the number of hours worked. Last week I worked 37.5 hr and my gross pay was 344.25 dollars. Write the variation equation and determine how much I will gross this week if I work 35 hr. What does the value of \(k\) represent in this case?

Graph each function using the Guidelines for Graphing Rational Functions, which is simply modified to include nonlinear asymptotes. Clearly label all intercepts and asymptotes and any additional points used to sketch the graph. $$Y_{1}=\frac{x^{3}-3 x+2}{x^{2}-9}$$

The surface area of a spherical cap is given by \(S=2 \pi r h,\) where \(r\) is the radius of the sphere and \(h\) is the perpendicular distance from the sphere's surface to the plane intersecting the sphere, forming the cap. The volume of the cap is \(V=\frac{1}{3} \pi h^{2}(3 r-h) .\) Similar to Exercise \(61,\) a formula can be found that will minimize the area of a cap that holds a specified volume. a. Solve the volume formula for the variable \(r\) b. Substitute the resulting expression for \(r\) into the surface area formula and simplify. The result is a formula for surface area given solely in terms of the volume \(V\) and the height \(h\). c. Assume the volume of the spherical cap is \(500 \mathrm{cm}^{3} .\) Use a graphing calculator to graph the resulting function on an appropriate window, and use the graph to find the height \(h\) that will result in a spherical cap with the smallest possible area, while still holding a volume of \(500 \mathrm{cm}^{3}\) d. Use this value of \(h\) and \(V=500 \mathrm{cm}^{3}\) to find the radius of the sphere.
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