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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. • - 121=-11 121 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=+/• 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 d. 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 < or > 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.