Choosing a Method of Solution Although you can use the quadratic formula to solve any equation, it is often much easier to factor or complete the square. The list below suggests when to use which method. SituationMethod to use

If an equation contains variables on both sides or variables in the denominator, then you must carefully organize your method for solving in order not to lose a root or gain a root. It is possible to lose a root by dividing both sides of an equation by a common factor.

Two ways to avoid losing a root are shown below. Both methods are correct. Method 1Method 2 If there is a factor common to both sides of the equation, remember to include as roots all values that make this factor zero. Bring all terms to one side of the equation and then solve.

Gaining a root Squaring both sides or multiplying by an expression may give you an extraneous root which satisfies the transformed equation but not the original one. Multiply each side by the LCD

Check: Undefined, so x = 2 is not a solution.

Ex. 1. The senior class has paid $200 to rent a roller skating rink for a fund raising party. Tickets for the party are $5 each. a.Express the net income as a function of the number of tickets sold. b.Graph the function. Identify the point at which the class begins to make a profit. Net income is the amount of money remaining after operating expenses are deducted from the amount of money earned.

Since n must be a nonnegative integer, the graph consists of discrete (isolated) points on a line. However, the line that contains the discrete points is often given as a sketch of the function. From these graphs you can see that n = 40 is a zero of the income function. Therefore, the class must sell more than 40 tickets to make a profit.

The domain of a function is the set of values for which the function is defined. You can think of the domain of a function as the set of input values. The set of output values is called the range of the function. A mathematical model is one or more functions, graphs, tables, equations, or inequalities that describe a real-world situation. Example 2 illustrates another model

Ex. 2. Suppose that it costs 50 cents for the first minute of a long distance telephone call and 20 cents for each additional minute or fraction thereof. Give a graphical model of the cost of a call lasting t minutes. It is important to realize that the cost of a call lasting 2 minutes and 12 seconds is the same as a 3 minute call. Likewise, a three and a half minute call costs the same as a 4 minute call. The function that models this cost is a step function, so named because its graph has steps.

Be aware that if you use the linear function, you will get overestimates of the cost.

Quadratic Functions and their Graphs If a graph has an axis of symmetry, then when you fold the graph along this axis, the two halves of the graph coincide. The graph of a quadratic function has a vertical axis of symmetry, or axis. The vertex of the parabola is the point where the axis of symmetry intersects the parabola.

To sketch the graph of Decide if it opens up or down and the number of x-intercepts Find the axis and vertex Find the x- and y-intercepts

Ex. 1. Sketch the parabola. Label the intercepts, axis of symmetry, and vertex. Does it open up or down? Find the axis To find the vertex, plug in 2 for x and determine the y-coordinate. Find the x and y-intercepts

Ex. 2. a.Find the vertex of the parabola by completing the square. b.Find the x- and y-intercepts a. Put in vertex form: where (h, k)is the vertex The vertex is (3, 22)

b. When x = 0, y = 4. So the y-intercept is (0, 4) To find the x-intercept, let y = 0

Ex. 3. Where does the line y = 3x + 5 intersect the parabola Set and solve for x Substitute these into y = 3x + 5 to get y = -7 and y = 8 So the intersection points are (-4, -7) and (1, 8) Graph to confirm your answer

Ex. 4. Find an equation of the function whose graph is a parabola with x-intercepts (3, 0) and (6, 0) and y-intercept (0, -2). If x = 3 and x = 6 are solutions of this equation, then (x – 3) and (x – 6) are factors of the equation so Use the y-intercept Graph to confirm