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Chapter 2: Problem 66

Find two quadratic functions, one that opens upward and one that opens downward, whose graphs have the given \(x\) -intercepts. (There are many correct answers.) (-5,0),(5,0)

Short Answer

Expert verified
The two quadratic functions that open upward and downward respectively, while having x-intercepts are (-5,0) and (5,0) are \(f_1(x) = x^2 - 25\) and \(f_2(x) = -x^2 + 25\).

Step by step solution

01

Identifying the zeros

The x-intercepts given in the problem are (-5,0) and (5,0), so the zeros of the functions are -5 and 5.
02

Choose the leading coefficients

Choose a leading coefficient a such that the first quadratic function opens upward and the second opens downward. Thus, it is possible to, for instance, choose \(a = 1\) for the upward-opening function, and \(a = -1\) for the downward-opening function.
03

Plug in the zeros and leading coefficients

Plug the zeros and the leading coefficients into the generic form \(f(x) = a(x-h)(x-k)\). The upward-opening function \(f(x) = 1\cdot(x - (-5))(x - 5)\) simplifies to \(f_1 (x) = (x+5)(x-5)\). The downward-opening function \(g(x) = -1\cdot(x - (-5))(x - 5)\) simplifies to \(f_2 (x) = - (x+5)(x-5)\).
04

Simplify the Formulas

The simplified form of the two quadratic functions are: \(f_1(x) = x^2 - 25\) and \(f_2(x) = -x^2 + 25\).
05

Confirm the solutions

By inspection we can confirm that \(f_1(x)\) opens upwards because its leading coefficient is positive, and that \(f_2(x)\) opens downwards because its leading coefficient is negative, and they both have x-intercepts at (-5,0) and (5,0).

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Key Concepts

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

Parabolas
A parabola is a symmetrical curve that's important in mathematics, especially in the context of quadratic functions. It is the graph of a quadratic equation, which is generally in the form of \( y = ax^2 + bx + c \). This equation represents a parabola in the Cartesian coordinate system.

Parabolas have some unique characteristics:
  • Symmetry: Parabolas are always symmetrical with respect to a vertical line known as the axis of symmetry.
  • Vertex: The highest or lowest point on a parabola is called the vertex. The vertex is always on the axis of symmetry.
In our example, the quadratic functions graph as parabolas. The direction the parabolas open (either upward or downward) is determined by the leading coefficient (\(a\)) of the quadratic function.

Understanding parabolas is essential for graphing any quadratic function and solving related problems.
x-intercepts
The x-intercepts of a graph are the points where the graph crosses the x-axis. For a quadratic function, these are also known as the roots or zeros. These points are critical because they can help us determine the function's equation.

Given in our exercise are x-intercepts at \((-5, 0)\) and \((5, 0)\). This tells us that the associated quadratic functions will be in the form \((x + 5)(x - 5)\), because when either \(x + 5\) or \(x - 5\) equals zero, the product will be zero, placing the points on the x-axis.
  • This can be expressed as two factors: \((x + 5)\) and \((x - 5)\).
  • These intercepts form the foundation of the equations we are working with.
Recognizing the x-intercepts forms the basis for writing the quadratic function that accurately represents the given function graphically.
Zeros
Zeros, like the x-intercepts, are the points where the graph of a quadratic function touches or crosses the x-axis. They are called zeros because when you substitute these values of x into the quadratic equation, the resulting value of y, or the function value, is zero.

In our quadratic equations, the zeros are at \(x = -5\) and \(x = 5\). These zeros correspond to the x-intercepts we've identified.
  • The zeros of a quadratic function are critical in determining the factors of the polynomial.
  • They indicate the solutions of the equation \(ax^2 + bx + c = 0\).
Understanding zeros helps in solving quadratic equations and constructing their graphs.
Leading Coefficients
The leading coefficient in a quadratic function is the coefficient of the \(x^2\) term when the function is in standard form, \(ax^2 + bx + c\). This coefficient significantly affects the shape and direction of the parabola.

For instance:
  • If the leading coefficient \(a\) is positive, the parabola opens upward. This configuration creates a U-shaped graph.
  • If \(a\) is negative, the parabola opens downward, resembling an inverted U.
In the exercise, we used a leading coefficient of \(1\) for the upward-opening function, resulting in \(f_1(x) = x^2 - 25\). For the downward-opening function, we used a leading coefficient of \(-1\), giving us \(f_2(x) = -x^2 + 25\).
Being mindful of the leading coefficients allows us to predict and control the opening direction of a parabola within a quadratic function.

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Most popular questions from this chapter

Use a graphing utility to graph the rational function. Give the domain of the function and identify any asymptotes. Then zoom out sufficiently far so that the graph appears as a line. Identify the line. \(g(x)=\frac{1+3 x^{2}-x^{3}}{x^{2}}\)

A driver averaged 50 miles per hour on the round trip between Akron, Ohio, and Columbus, Ohio, 100 miles away. The average speeds for going and returning were \(x\) and \(y\) miles per hour, respectively. (a) Show that \(y=\frac{25 x}{x-25}\). (b) Determine the vertical and horizontal asymptotes of the graph of the function. (c) Use a graphing utility to graph the function. (d) Complete the table. $$\begin{array}{|l|l|l|l|l|l|l|l|}\hline x & 30 & 35 & 40 & 45 & 50 & 55 & 60 \\\\\hline y & & & & & & & \\\\\hline\end{array}$$ (e) Are the results in the table what you expected? Explain. (f) Is it possible to average 20 miles per hour in one direction and still average 50 miles per hour on the round trip? Explain.

Solve the inequality and graph the solution on the real number line. \(x^{2}<9\)

(a) find the interval(s) for \(b\) such that the equation has at least one real solution and (b) write a conjecture about the interval(s) based on the values of the coefficients. \(3 x^{2}+b x+10=0\)

Solve the inequality and write the solution set in interval notation. \(x^{4}(x-3) \leq 0\)
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