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Chapter 2: Problem 70

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.) $$\left(-\frac{5}{2}, 0\right),(2,0)$$

Short Answer

Expert verified
The upward opening quadratic function is \(f(x) = (x + \frac{5}{2})(x - 2)\) and the downward opening quadratic function is \(g(x) = -(x + \frac{5}{2})(x - 2)\)

Step by step solution

01

Formulate the Quadratic Functions

Using the general form \(f(x) = a(x - r)(x - s)\), replace \(r\) and \(s\) with the given \(x\)-intercepts. Here, \(r = -\frac{5}{2}\) and \(s = 2\). Therefore, the quadratic function is \(f(x) = a(x - (-\frac{5}{2}))(x - 2)\), which simplifies to \(f(x) = a(x + \frac{5}{2})(x - 2)\).
02

Define the Upward Opening Function

To ensure the quadratic function opens upward, \(a\) should be positive. Let's choose \(a = 1\) for simplicity. Substituting \(a\) into the equation gives the upward opening quadratic function: \(f(x) = (x + \frac{5}{2})(x - 2)\).
03

Define the Downward Opening Function

Similarly, to ensure the quadratic function opens downward, \(a\) should be negative. Let's choose \(a = -1\) for simplicity. Substituting \(a\) into the equation gives the downward opening quadratic function: \(g(x) = -(x + \frac{5}{2})(x - 2)\).

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Most popular questions from this chapter

Write the polynomial as the product of linear factors and list all the zeros of the function. $$g(x)=x^{4}-4 x^{3}+8 x^{2}-16 x+16$$

Decide whether the statement is true or false. Justify your answer. It is possible for a third-degree polynomial function with integer coefficients to have no real zeros.

Prove that the complex conjugate of the sum of two complex numbers \(a_{1}+b_{1} i\) and \(a_{2}+b_{2} i\) is the sum of their complex conjugates.

Use Descartes's Rule of Signs to determine the possible numbers of positive and negative real zeros of the function. $$f(x)=4 x^{3}-3 x^{2}+2 x-1$$

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