Chapter 10 Quadratic Equations and Functions Section 10-1 Exploring Quadratic Graphs A quadratic function is a function of degree 2 The graph of a quadratic function is a parabola The line that cuts a parabola in half is the axis of symmetry The highest or lowest point (depending on the direction of the parabola) is called the vertex → it is the point where the graph changes direction

Parabolas have reflectional symmetry over the axis of symmetry (they are exactly the same on both sides) To graph a parabola: make a table of values until you have enough to plot and see the shape The coefficient with the x² term determines the shape of the graph If the coefficient is negative, the parabola is up-side-down (it has a maximum) If the coefficient is positive, the parabola is right-side-up (it has a minimum)

If the absolute value of the coefficient is 0 < n < 1, the shape will be wider than the parent function (y = x²) If the absolute value of the coefficient is n > 1 then the shape will be skinnier than the parent function Any number added to or subtracted from the x² term will shift the vertex Quadratic formulas are used to model the height of a launched object over time

Graph Ex1. y = 2x² + 4 Ex2. y = -¾x² – 5

Section 10-2 Quadratic Functions The graph of y = ax² + bx + c, where a ≠ 0, has the line as its axis of symmetry The x-coordinate of the vertex is To graph a quadratic of the form y = ax² + bx + c: A) find the axis of symmetry B) find the vertex by plugging in your answer from part A in the equation for x (this will give you the coordinates of the vertex)

Graphing continued C) Plug in other values for x and find the corresponding y-values (remember that since the graph has reflectional symmetry, the point equidistant on the opposite side of the axis of symmetry is also a point on the graph – see example 1 on page 518) D) Connect To graph a quadratic inequality, graph the parabola and then shade (< and < shade below, > and > shade above)

Ex1. y = 2x² + 4x – 3 A) Find the equation for the axis of symmetry B) Find the coordinates of the vertex C) Find 4 other points on the parabola D) Graph

Ex2. Graph y > -x² + 6x – 5

Section 10-3 Finding and Estimating Square Roots means the positive square root of x means the negative square root of x You should use the ± symbol to indicate the positive and negative answer Ex1. Simplify a) b) c) d)

Rational numbers end or repeat in decimal form Irrational numbers do not end or repeat in decimal form (pi) Ex2. List the first 15 perfect squares (know these!!) Ex3. Between what two consecutive integers is ? Ex4. Find the square roots of

Section 10-4 Solving Quadratic Equations One way to solve a quadratic equation is by graphing Graph the parabola and all of the places where the graph crosses the x-axis are the solutions (there could be one, none, or two) If in the equation y = ax² + bx + c, b = 0, then you can solve by using square roots (simply solve for x)

Ex1. Solve by graphing x² – 9 = 0 Ex2. Solve by finding the square roots. n² – 225 = 0

Review 10-1 through 10-4 10 – 1: Find the axis of symmetry and vertex for y = -6x² + 8 10 – 2: Find the axis of symmetry and vertex for y = 3x² + 6x – 5 (be able to graph too) 10 – 3: Simplify each of the following a) b) c) 10 – 4: Solve. 81x² = 169

Section 10-5 Factoring to Solve Quadratic Equations For every real number a and b, if ab = 0, then a = 0 or b = 0 By this property, you can solve to find the zeros (the solutions) of a quadratic formula by factoring Factor and solve Ex1. x² – 5x – 24 = 0 Ex2. x² – 9x = -20

Section 10-6 Completing the Square Some of the other methods we have discussed for solve quadratics only works for certain types, completing the square works EVERY time Completing the square involves adding the same number to both sides of an equation (which changes nothing) in order to make one side a perfect square trinomial

To complete the square, rewrite the equation so it is in the form x² + bx = c Add (½b)² to both sides Rewrite the left side as a binomial squared and simplify the right side Then to solve, find the square root of both sides (two possibilities on the right) and solve for x The directions should tell you when to round and when to give exact answers (if they don’t, give an exact answer)

Solve by completing the square. Ex1. x² + 5x – 50 = 0 Ex2. If there is a coefficient with the x² term, you will need to divide every coefficient in the equation by that number to eliminate it (the coefficient with x² must be 1) Ex3. 3x² + 6x – 24 = 0

Section 10-7 Using the Quadratic Formula The quadratic formula is another way to solve quadratic functions (it also works for all types) x = You must memorize this formula! Notice the ± before the square root (this is because you get 1, 0, or 2 answers)

When using this formula, if you are going to put it all into your calculator at once, be VERY careful with parentheses! It is safer to put in small amounts at a time to avoid mistakes! The radicand is what is underneath the radical sign (in this case b² – 4ac) Read the purple box on the bottom of page 549 to know when to use which method of solving (when it is your choice)

Solve using the quadratic formula Solve using the quadratic formula. Round to the nearest hundredth where necessary. Ex1. x² + 2 = -3x Ex2. 3x² + 4x – 8 = 0 Ex3. x² – 4x = 117

Section 10-8 Using the Discriminant The radicand of the quadratic formula is called the discriminant (b² – 4ac) The discriminant tells you how many answers there will be If the discriminant is positive, there are two answers If the discriminant is zero, there is one answer If the discriminant is negative, there are no real answers

This is shown graphically on pages 554 and 555 This is shown algebraically in the orange box on page 555 Ex1. Determine the number of solutions to 6x² + 3x – 5 = 0 Remember that the number of solutions is also the number of x-intercepts (since the intercepts are the solutions) Ex2. Determine the number of solutions to 2x² – 3x + 8 = 0

Section 10-9 Choosing a Linear, Quadratic, or Exponential Model One way to determine which model is appropriate if you are only given a set of points is to graph those points and see what shape they make There is another way that does not involve graphing Open your book to page 560, we are going to see another way to determine

Once you have determined which model is best, you can use the equation for that model and the information given to write an equation for the situation Which kind of data best models the data below? Write an equation to model the data. Ex1. Ex2. x y 1 1.4 2 5.6 3 12.6 4 22.4 x y -1 4 2 1 .5 3 .25