8/24/2015 V. J. Motto 1 Chapter 2: Review – Quadratic Function V. J. Motto M110 Modeling with Elementary Functions

Graphing a Quadratic Consider the quadratic is y = x 2. Three points will almost certainly not be enough points for graphing a quadratic, at least not until you are very experienced. 8/24/2015V. J. Motto 2

General form y = ax 2 + bx + c. The general form of a quadratic is y = ax 2 + bx + c. For graphing, the leading coefficient "a" indicates how "fat" or how "skinny" the parabola will be. 8/24/2015V. J. Motto 3

Which way is up? There is a simple, if slightly “silly", way to remember the difference between right-side-up parabolas and upside-down parabolas 8/24/2015V. J. Motto 4

Form y = a(x – h) 2 + k If the quadratic is written in the form y = a(x – h) 2 + k, then the vertex is the point (h, k). This makes sense, if you think about it. The squared part is always positive (for a right-side-up parabola), unless it's zero. So you'll always have that fixed value k, and then you'll always adding something to it to make y bigger, unless of course the squared part is zero. So the smallest y can possibly be is y = k, and this smallest value will happen when the squared part, x – h, equals zero. And the squared part is zero when x – h = 0, or when x = h. The same reasoning works, with k being the largest value and the squared part always subtracting from it, for upside-down parabolas. 8/24/2015V. J. Motto 5

Let’s Convert some Place the quadratic in vertex-form: y = x 2 – 6x + 7 8/24/2015V. J. Motto 6

Vertex Formula An alternate way of finding the vertex (the maximum or minimum points include: The formula: Use your calculator 8/24/2015V. J. Motto 7

Let’s use the calculator We will use the calculator to sketch the graph of y = x 2 – 6x + 7 and find the vertex. From looking at the graph we see that we should give the following: Vertex (maximum, minimum) y-intercept x-intercepts or zeros 8/24/2015V. J. Motto 8

8/24/2015V. J. Motto 9 Finding Zeros by Factoring You should recall from you study of Elementary Algebra Foundations the Zero Product Property. It stated that if ab = 0 then either a = 0 or b = 0. We can use this property to solve quadratic equations

8/24/2015V. J. Motto 10 Example 1: Subtract 9 from both sides. Factor. Zero Product Property Solve Solve by Factoring

8/24/2015V. J. Motto 11 Square Root Property If b is a real number and if a 2 = b, then

8/24/2015V. J. Motto 12 Example 2: Use the square root property to solve.

8/24/2015V. J. Motto 13 Example 3: Use the square root property to solve.

8/24/2015V. J. Motto 14 Perfect Square Trinomials Perfect Square Trinomial Factored Form Notice that for each perfect square trinomial, the constant term of the trinomial is the square of half the coefficient of the x-term provided That the coefficient of the squared term is 1.

8/24/2015V. J. Motto 15 Perfect Square Trinomials The process of writing a quadratic equation so that one side Is a perfect square trinomial is called completing the square.

8/24/2015V. J. Motto 16 Example 4: Solve by completing the square

8/24/2015V. J. Motto 17 Example 4 (continued)

8/24/2015V. J. Motto 18 Example 5:

8/24/2015V. J. Motto 19 Example 5 (continued)

8/24/2015V. J. Motto 20 The Quadratic Formula Only a few quadratic equations can be solved by factoring. However, any quadratic equation can be solved by completing the square. Since the same sequence of steps is repeated each time we completed the square, let’s generalize the solution into a formula!

8/24/2015V. J. Motto 21 The Development We start with the general quadratic equation: with a, b and c real numbers and a ≠ 0. Divide both sides by a. Subtract from both sides.

8/24/2015V. J. Motto 22 The Development -- 2 Using the coefficient of the x term, we find the expression that will complete the square. We add this expression to both sides of the equation.

8/24/2015V. J. Motto 23 The Development Find a common denominator On the right side. Do the multiplication. Simplify the right side.

8/24/2015V. J. Motto 24 The Development Factor the perfect square trinomial on the left Take the square root of both sides of the equation Simplify the radicals Solve the absolute value equation.

8/24/2015V. J. Motto 25 The Development Solve for x. Simplify the expression. This equation identifies the solutions of the general Quadratic equation in standard form.

8/24/2015V. J. Motto 26 Comments b 2 -4ac is called the discriminant. If b 2 -4ac ≥ 0 the solutions are real. If b 2 -4ac < 0, the solution are not real. If b 2 -4ac is perfect square root, the solutions are integers! (And you could have used factoring to find the solution.)

8/24/2015V. J. Motto 27 Example 1 Solve First, write the equation in standard form. Identify the coefficients of the quadratic equation and calculate the value of the discriminant. What does this value tell you about the solution?

8/24/2015V. J. Motto 28 Example 1 (continued) Substitute the values into the quadratic formula. The solutions are:

Using the calculator We will use the calculator to find the zeroes for y = x 2 – 6x /24/2015V. J. Motto 29

8/24/2015V. J. Motto 30 Group Work Sketch the graph for identifying all the important points: