High School Algebra I Unlocked (2016)
Chapter 8. Quadratic Functions
By the end of this chapter, you will be able to:
•Explain the key features of quadratic functions
•Determine the solutions to a quadratic function by factoring, completing the square, or using the quadratic formula
•Use the discriminant to determine whether a quadratic function has real or complex solutions
•Find the intercepts, minimum and maximum, and axis of symmetry of a quadratic function
•Find the domain and range, intervals of increase and decrease, and end behavior of a quadratic function
•Graph quadratic functions
Lesson 8.1. Introduction to Quadratic Functions
In Chapter 6, we discussed quadratic equations—equations with a degree of 2 that are written in standard form as ax2 + bx + c = 0. Quadratic equations are used to find the specific value of a variable and, therefore, are always equal to a number. A quadratic equation may be factored in the form (x + m)(x + n), which allows us to determine that the roots, or solutions, of the quadratic equation are x = −m or x = −n.
In this chapter we are going to expand upon our knowledge of quadratics by discussing the purpose and characteristics of quadratic functions. Unlike quadratic equations, quadratic functions are written in standard form as f(x) = ax2 +bx + c and are always set equal to f(x) or y. A quadratic function is the algebraic representation of the path of a parabola, the symmetrical curve produced by a quadratic equation.
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Now, if you are scratching your head at this point, thinking, “Quadratic functions sound exactly like quadratic equations,” you aren’t crazy. Quadratic functions are approached in much the same way as quadratic equations. The main difference is that a quadratic function is set equal to f(x), or y, which means that you are not solving for a single value. Rather, a function allows you to determine the output for multiple inputs, where the output is dependent upon the input.
Imagine that you have decided to sell miniature unicorns on a website. Being the entrepreneur that you are, you would like to know how much profit you can expect to make after a certain period of time. Your amazing accountant tells you that the profit you will make can be found using the function p(x) = −.005x2 + 20x − 400, which accounts for the number of miniature unicorns sold, the amount earned in sales, and the costs of running your business. Using this information, you can find your profit for any number of miniature unicorns sold, as shown on the following graph:
Using the graph of the function p(x) = −.005x2 + 20x − 400, you can determine the profit of your business for all prices of miniature unicorns. Based on the graph, you can see that your profit would be $0 at p(20) and p(3980), or a sale price of $20 and $3,980. Conversely, you would achieve a maximum profit of $19,600 at p(2000), which, due to the symmetrical nature of parabolas, can be found when x is halfway between the zeros; (20 + 3,980)/2 = 2,000. Furthermore, we can also find your profit for any sale price in between—which is the real power of functions.
RATIONAL AND IRRATIONAL NUMBERS
Before we get into some of the key features of quadratic functions, let’s talk a bit about the world of rational and irrational numbers. A rational number is one that can be expressed as a ratio of two integers, while an irrational number, such as pi, cannot be expressed as a ratio. In Chapter 6, we solved quadratic equations that had one or two real solutions, and discussed how some quadratic equations have complex solutions—or solutions that include the imaginary number i. However, you don’t need to worry about complex solutions until Algebra II.
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While you probably will not work with complex solutions in Algebra I, you will need to determine how may solutions a quadratic equation has by finding the discriminant, or the value of b2 − 4ac.
• If the discriminant is positive, the quadratic function will have two real solutions.
• If the discriminant is zero, the quadratic function will have one real solution.
• If the discriminant is negative, the quadratic function will have two complex solutions, or solutions that include the imaginary number i.
Now that you have a little review under your belt, let’s dive into quadratic functions!