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Chapter 1: Problem 57

Graph the function and determine the interval(s) for which \(f(x) \geq 0\) . \(f(x)=9-x^{2}\)

Short Answer

Expert verified
The function \(f(x)\) is greater than or equal to zero for \(x\) in the interval \([-3,3]\).

Step by step solution

01

Find the zeros of the function.

The zeros of a function are the x-values for which \(f(x) = 0\). This is found by solving the equation \(f(x) = 0\), in this case \(9-x^{2} = 0\). Solving for \(x\) yields \(x = \pm \sqrt{9}\), or \(x = \pm 3\). These are the zeros of the function.
02

Determine on which intervals the function is positive.

The function equals zero at \(x = -3\) and \(x = 3\) and it is a downward parabola. Hence the function \(f(x)\) is greater than or equal to zero for \(x\) in the interval \([-3,3]\).
03

Graph the function

Take some key points, typically the roots calculated in step 1 and few points on either side of the roots, and plot them. This will give a curve opening downward and crossing the x-axis at \(x=-3\) and \(x=3\).

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Key Concepts

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

Graphing Quadratics
Quadratic functions are often graphed as parabolas, which are U-shaped curves. A quadratic function is generally expressed as \( f(x) = ax^2 + bx + c \). The graph of a quadratic function is a parabola that can open upwards or downwards.
  • If \( a > 0 \), the parabola opens upwards.
  • If \( a < 0 \), the parabola opens downwards.
In the example \( f(x) = 9 - x^2 \), the coefficient of \( x^2 \) is negative, meaning the parabola opens downward. To graph this function:
  • Identify the vertex, which for the function \( f(x) = ax^2 + bx + c \) is given by the point \( (0, c) \) when there's no linear term. Hence, the vertex here is \( (0, 9) \).
  • The parabola is symmetric around the y-axis because the quadratic term does not include a linear term (i.e., \( b = 0 \)).
  • Plot the zeros and a few additional points to visualize the curve.
Plotting these points helps to accurately sketch the parabolic shape and understand its behavior in different intervals.
Finding Zeros of a Function
The zeros of a function are the x-values where the function equals zero. For quadratic functions, finding the zeros helps determine where the graph crosses the x-axis.
To find the zeros of \( f(x) = 9 - x^2 \), set the equation to zero and solve: \[ 9 - x^2 = 0 \]\[ x^2 = 9 \]\[ x = \pm 3 \]These solutions \( x = 3 \) and \( x = -3 \) are the points where the graph intersects the x-axis. Knowing the zeros is crucial as they form the boundary of intervals where the function is positive or negative.
Parabolas
Parabolas have a distinct shape characterized by their symmetry and direction. They can either open upwards or downwards, determined by the sign of their highest degree term.
Different elements define the nature of a parabola:
  • Vertex: The peak or lowest point, located at the axis of symmetry.
  • Axis of Symmetry: A vertical line splitting the parabola into two equal mirror-image halves. For \( f(x) = 9 - x^2 \), this is the y-axis \( x = 0 \).
  • Focus and Directrix: Points that give parabolas their shape, but for basic graphing, we focus on the vertex and symmetry.
Understanding these components helps in sketching a parabola accurately and predicting its behavior across its graph. Parabolas like \( f(x) = 9 - x^2 \) that open downwards have a maximum point, reinforced by negative leading coefficients.
Function Intervals
The concept of intervals in a quadratic function refers to the x-range where the function displays specific behavior, like being positive, negative, or zero.
To determine intervals where \( f(x) \geq 0 \):
  • Use the zeros \( x = -3 \) and \( x = 3 \), which divide the function into different sections.
  • Since the parabola \( f(x) = 9 - x^2 \) opens downward, \( f(x) \) will be above the x-axis in the range between these zeros.
  • This means \( f(x) \geq 0 \) on the closed interval \([-3, 3]\), where the function is zero at \( x = -3 \) and \( x = 3 \) and positive in between.
By understanding the intervals and their boundaries, one can better interpret the overall behavior and graphical representation of a quadratic function.

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Most popular questions from this chapter

Brain Weight The average weight of a male child's brain is 970 grams at age 1 and 1270 grams at age 3 . (a) Assuming that the relationship between brain weight \(y\) and age \(t\) is linear, write a linear model for the data. (b) What is the slope and what does it tell you about brain weight? (c) Use your model to estimate the average brain weight at age \(2 .\) (d) Use your school's library, the Internet, or some other reference source to find the actual average brain weight at age \(2 .\) How close was your estimate? (e) Do you think your model could be used to determine the average brain weight of an adult? Explain.

Use a graphing utility to graph each function. Write a paragraph describing any similarities and differences you observe among the graphs. (a) \(y=x\) (b) \(y=x^{2}\) (c) \(y=x^{3}\) (d) \(y=x^{4}\) (e) \(y=x^{5}\) (f) \(y=x^{6}\)

Think About It The function \(f(x)=k\left(2-x-x^{3}\right)\) has an inverse function, and \(f^{-1}(3)=-2 .\) Find \(k\)

Finding Parallel and Perpendicular, write equations of the lines through the given point (a) parallel to and (b) perpendicular to the given line. $$6 x+2 y=9, \quad(-3.9,-1.4)$$

Matching and Determining Constants In Exercises \(85-88\) , match the data with one of the following functions $$ f(x)=c x, g(x)=c x^{2}, h(x)=c \sqrt{|x|}, \text { and } r(x)=\frac{c}{x} $$ and determine the value of the constant \(c\) that will make the function fit the data in the table. $$ \begin{array}{|c|c|c|c|c|c|}\hline x & {-4} & {-1} & {0} & {1} & {4} \\\ \hline y & {-32} & {-2} & {0} & {-2} & {-32} \\ \hline\end{array} $$
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