Chapter 9 Quadratic Equations and Functions By: Courtney Popp & Kayla Kloss

9.1-Solving Quadratic Equations by Finding Square Roots All positive real numbers have two square roots: a positive and negative square root. 3^2=9 then 3 is a square root of 9. A radicand is the number or expression inside a radical symbol. Perfect squares are numbers whose square roots are integers or quotients of integers. An irrational number is a number that cannot be written as the quotient of two integers = is a perfect square: 11^2=121

9.1 Continued A quadratic equation is an equation that can be written in the following standard form. ax^2+bx+c=0, where a 0 In standard form, a is the leading coefficient. Solving x^2=d by finding square roots: If d>0, then x=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 real solution. x^2=4 has two solutions: x= -2, +2 x^2=0 has one solution: x= 0 x^2= -1 has no real solution Falling Object Model: h= -16t^2+s h=height, t=time, s=initial height s=32; h= -16t^2+32

9.2-Simplifying Radicals Product Property: the square root of a product equals the product of the square roots of the factors. ab= a x b when a and b are positive numbers. 4 x 100= 4 x 100 Quotient Property: the square root of a quotient equals the quotient of the square roots of the numerator and denominator. a/b = a/ b when a and b are positive numbers. 9/25 = 9/ 25 An expression with radicals is in simplest form if the following are true: No perfect square factors other than 1 are in the radicand. No fractions are in the radicand. No radicals appear in the denominator of a fraction.

9.3-Graphing Quadratic Functions Every quadratic function has a U-shaped graph called a parabola. If the leading coefficient a is positive the parabola opens up, if it’s negative the parabola opens down. The vertex is the lowest point of a parabola that opens up and the highest point of a parabola that opens down. The line passing through the vertex that divides the parabola into two symmetric parts is called the axis of symmetry. Graph of a Quadratic Function: the graph of y=ax^2+bx+c is a parabola. The vertex has an x-coordinate –b/2a. The axis of symmetry is the vertical line x= -b/2a. To Graph: Find the x-coordinate of the vertex. Make a table of values, using x-values to the left and right of the vertex. Plot the points and connect them with a smooth curve to form a parabola.

9.4-Solving Quadratic Equations by Graphing Step 1: Write the equation in the form ax^2+bx+c=0 Step 2: Write the related function y=ax^2+bx+c Step 3: Sketch the graph of the function y=ax^2+bx+c The solutions of ax^2+bx+c=0 are the x-intercepts.

9.5-Solving Quadratic Equations by the Quadratic Formula Quadratic Formula: Use this formula to solve quadratic equation: ax^2+bx+c=0

9.5 Continued Vertical Motion Models: Object is dropped: h= -16t^2+s Object is thrown: h= -16t^2+vt+s h=height (ft), t=time in motion (sec) s=initial height (ft), v=initial velocity (ft/sec) Example: s=200ft, v= -30ft/sec (the object thrown) h= -16t^2+( -30)t+200 Substitute 0 for h, write in standard form. t= -(-30)+/- (-30)^2-4(-16)(200) over 2(-16) Simplify: t=30+/- 13,700 over -32 Solutions: t=2.72 or -4.60

9.6-Applications of the Discriminant :b^2-4ac=discriminant The discriminant is used to find the number of solutions of a quadratic equation. Number of solutions of a quadratic equation: If discriminant is positive, then 2 solutions. If discriminant is 0, then 1 solution. If discriminant is negative, then no real solutions. Example: x^2-3x-4=0 b^2-4ac=(-3)^2-4(1)(-4) =9+16 =25, discriminant is positive. 2 solutions.

9.7- Graphing Quadratic Inequalities If quadratic inequality has then it will have a dotted parabola. If inequality has or than it will have a solid lined parabola. Check a point somewhere in the parabola; if it’s a solution, shade its region. If not, shade the other region.

9.8-Comparing Linear, Exponential, and Quadratic Models. Three basic models: Linear= y=mx+b Exponential= y=c(1+/-r)^t Quadratic= y=ax^2+bx+c

9.8 Continued (0, 3) (8, 3) (-4, -1) (4, 4) (-6, -3) (10, 1) Plot points on graph See what shape it gives you, decide between linear, exponential, and quadratic models.