Name Chapter 5 Smartboard Notes 10.1 Graph ax 2 + c Learning Outcome To graph simple quadratic functions Quadratic function A non linear function that can be written in the standard form y = ax 2 + bx + c where a 0 Parabola A U shaped graph of a quadratic function Parent quadratic function The most basic quadratic function y = x 2 Vertex The lowest or highest point on a parabola Axis of Symmetry The line that passes through the and divides the parabola into two symmetric parts Comparing the graph of y = ax 2 if a > 1 then vertical stretch if 0 < a < 1, then vertical shrink if a < - 1 then vertical stretch (reflection in the x-axis) if -1 < a < 0, then vertical shrink (reflection in the x-axis) When y = ax 2, a < 0 open DOWN When y = ax 2, a > 0 open UP Graph y = ax 2 where a < l Graph y =1 x 2 Compare the graph with the graph of y = x 2 2
Graph y = 5x 2. Compare the graph with the graph of y = x 2. Graph y = x 2 2. Compare the graph with the graph of y = x 2. Graph y = x 2 6. Compare the graph with the graph of y = x 2. 10.2 Graph ax 2 + bx +c Learning Outcome To graph general quadratic functions Quadratic function A non linear function that can be written in the standard form y = ax 2 + bx + c where a 0 Minimum value When a > 0, the minimum value of the function y = ax 2 + bx + c is the y-coordinate of the. Maximum value When a< 0, the maximum value of the function y= ax 2 + bx+ c is the y-coordinate of the. Properties of a quadratic function graph The graph of y = ax 2 + bx + c is a parabola that: opens up if a > 0 and opens down if a < 0 is narrower than the graph of y = x 2 if a > 1 and wider if a < 1 has an of x = - b 2a has a with an x-coordinate of x = - b 2a has a of c. So, the point (0,c) is on the parabola.
Minimum value When a > 0 Maximum value When a< 0 x = - b 2a (0,c) (0,c) x = - b 2a f(x) = 5x 2 20x + 17 domain range x-intercept f(x) = -.5x 2 + 6x + 8 domain range x-intercept
f(x) = -x 2 + 4x 1 domain range x-intercept f(x) = 3x 2 + 2x 5 domain range x-intercept f(x) = -3x 2 + 9x 8 domain range x-intercept
10.3 Solve Quadratic Equations by Graphing Learning Outcome To solve quadratic functions by graphing Quadratic function A non linear function that can be written in the standard form y = ax 2 + bx + c where a 0 Number of solutions of a quadratic equation A quadratic equation has two solutions if the graph of its related function has two x-intercepts A quadratic equation has one solution if the graph of its related function has one x-intercept A quadratic equation has no solution if the graph of its related function has no x-intercepts Solve x 2 + 2x = -8 by graphing (change to standard form) Solve equation (using FOIL) x-intercepts Solve x 2 4x = -4 by graphing Solve equation (using FOIL) x-intercepts
Solve x 2 + 8 = 2x by graphing Solve equation (using FOIL) x-intercepts Solve x 2 6 = -5x by graphing Solve equation (using FOIL) x-intercepts Find the zeros of f(x) = -x 2 8x 7 (then graph) Zeros (x-intercepts)
Find the zeros of f(x) = x 2 11x +18 (then graph) Zeros (x-intercepts) 10.4 Use Square Roots to Solve Quadratic Equations Learning Outcome To solve quadratic functions by finding square roots Solving x 2 = d by taking square roots If d > 0, then x 2 = d has two solutions: x = ± d If d = 0, then x 2 = d has one solution: x = 0 If d < 0, then x 2 = d has no solution Solve the equations z 2 5 = 4 r 2 + 7 = 4 3x 2 = 108 25k 2 = 9 t 2 + 17 = 17 81p 2 = 4 Approximate the solution 4x 2 + 3 = 23 2x 2 7 = 9 6g 2 + 1 = 19
5(x + 1) 2 = 30 4(a 3) 2 = 32 check your solutions using graphing calculator 10.5 Solve Quadratic Equations by Completing the Sqaure Learning Outcome To solve quadratic functions by completing the square Completing the square To complete the square, a constant c is added to an expression of the form x 2 + bx to form a perfect square trinomial. To complete the square for the expression x 2 + bx, add the square of half the coefficient of the term bx. Complete the square (easy) x 2 5x + c x 2 + 8x + c x 2 + 7x + c x 2 6x + c Complete the square (harder) Solve by completing the square MUST be in the form ax 2 + bx = c t 2 + 6t = 5 ( ) 2 = -5 + 2 Solve by completing the square MUST be in the form ax 2 + bx = c (where a = 1) 4m 2 16m + 8 = 0 ( ) 2 = + 2 5s 2 + 60s + 125 = 0 ( ) 2 = + 2
6k 2 60k + 12 = 0 ( ) 2 = + 2 10.6 Solve Quadratic Equations by Quadratic Formula Learning Outcome To solve quadratic functions by quadratic formula Quadratic formula The quadratic formula gives the solutions of any quadratic equation in standard form. Quadratic formula -b ± b 2-4ac 2a Solve using quadratic formula 2x 2 5 = 3x A crabbing net is thrown from a bridge, which is 35 feet above the water. If the net s initial velocity is 10 feet per second, how long will it take the net to hit the water? h = 16t 2 + vt + s (Remember the vertical motion model) Solve using quadratic formula 2x 2 + x = 3
Why would you use quadratic equation to solve? 4x 2 36 = 0 What method would you use to solve this equation? x 2 + 8x = 9 FOIL, Completing the square, or -b ± b 2-4ac 2a 10.7 Interpret the Discriminant Learning Outcome To use the value of the discriminant Discriminant In the quadratic formula, the expression b 2 4ac is called the discriminant of the associated equation ax 2 + bx + c = 0. Using the discriminant Value of discriminant b 2 4ac > 0 Number of solutions TWO TWO ONE NONE b 2 4ac = 0 ONE b 2 4ac < 0 NONE TWO ONE NONE Use the discriminant
Equation Discriminant ax + bx + c = 0 b 2 4ac x 2 3x 2 = 0 ( 3) 2 4( 1 )( 2 ) = 17 3x 2 + 2 = 0 0 2 4( 3 )( 2 ) = 24 2x 2 + 8x + 8 = 0 8 2 4(2)( 8 ) = 0 Use the discriminant to find the number of solutions 2x 2 + 4x = 2 3x 2 + 7x = 5 x 2 + 2x = 1 x 2 9 = 6x Find the number of x-intercepts (use discriminant) y = x 2 + 3x + 4 y = x 2 + 3x 3 y = x 2 4x + 4 y = 2x 2 4x + 2 10.8 Compare Linear, Exponential, and Quadratic Models Learning Outcome To compare linear, exponential, and quadratic Models Linear Function y = mx + b Exponential Function y = ab x Quadratic Function y = ax 2 + bx + c
Tell whether the ordered pair represents a linear, exponential or quadratic function by finding a pattern -2 7-1 1 0-1 1 1 2 7-2 4-1 2 0 1 1 1/2 2 1/4-2 5-1 3 0 1 1-1 2-3 Rule? Rule? Rule?
Tell whether the ordered pair represents a linear, exponential or quadratic function by finding the equation (rule) -2-12 -1-8 0-4 1 0 2 4-2.25-1.5 0 1 1 2 2 4-2 -1-1 1 0 3 1 5 2 7 Rule? Rule? Rule?
Tell whether the ordered pair represents a linear, exponential or quadratic function by finding the rule -2 3-1.75 0 0 1.75 2 3-2 32-1 8 0 2 1.5 2.125-2 1/9-1 1/3 0 1 1 3 2 9 Rule? Rule? Rule?
Tell whether the ordered pair represents a linear, exponential or quadratic function by finding the rule (harder) -2 0-1 -2 0 0 1 6 2 16-2 -3-1 -6 0-5 1 0 2 9-2 23-1 3 0-7 1-7 2 3 Rule? Rule? Rule?
Tell whether the ordered pair represents a linear, exponential or quadratic function by finding the rule (harder) -2 2/9-1 6/9 0 2 1 6 2 18-2 16-1 8 0 4 1 2 2 1-2 1/3-1 1 0 3 1 9 2 27 Rule? Rule? Rule?