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Exercises Polynomials and factoring

Example 4 from Algebra and Trigonometry, 5.1 Quadratic Functions

Finding the Domain and Range of a Quadratic Function

Find the domain and range of $f (x) = - 5 x^{2} + 9 x - 1.$

Work it out on paper first. Then open the solution one step at a time, and stop as soon as you can finish on your own.

Solution
As with any quadratic function, the domain is all real numbers.

Because $a$ is negative, the parabola opens downward and has a maximum value. We need to determine the maximum value. We can begin by finding the $x -$ value of the vertex.

$\begin{matrix} h & = & - \frac{b}{2 a} \\ = & - \frac{9}{2 (−5)} \\ = & \frac{9}{10} \end{matrix}$

The maximum value is given by $f (h) .$

$\begin{matrix} f (\frac{9}{10}) & = & -5 {(\frac{9}{10})}^{2} + 9 (\frac{9}{10}) - 1 \\ = & \frac{61}{20} \end{matrix}$
The range is $f (x) \leq \frac{61}{20},$ or $(- \infty, \frac{61}{20}] .$

Your turn

The book's Try It right after this example: same idea, new numbers. Only the answer is given.

Find the domain and range of $f (x) = 2 {(x - \frac{4}{7})}^{2} + \frac{8}{11} .$

How did it go?

PreviousFinding the Vertex of a Quadratic Function Next Finding the Maximum Value of a Quadratic Function

From Algebra and Trigonometry (1st edition) by OpenStax (Rice University), section 5.1, licensed under CC BY 4.0. Original at openstax.org. We split the solution into steps; the words, numbers and figures are the book's. OpenStax doesn't endorse this site.