7.1 Quadratic Equations Quadratic Equation: Zero-Factor Property: If a and b are real numbers and if ab=0 then either a = 0 or b = 0

7.1 Quadratic Equations Solving a Quadratic Equation by factoring 1.Write in standard form – all terms on one side of equal sign and zero on the other 2.Factor (completely) 3.Set all factors equal to zero and solve the resulting equations 4.(if time available) check your answers in the original equation

7.1 Quadratic Equations Example:

7.1 Quadratic Equations If 2 resistors are in series the resistance is 8 ohms and in parallel the resistance is 1.5 ohm. What are the resistances?

7.2 Completing the Square Square Root Property of Equations: If k is a positive number and if a 2 = k, then and the solution set is:

7.2 Completing the Square Example:

7.2 Completing the Square Example of completing the square:

7.2 Completing the Square Completing the Square (ax 2 + bx + c = 0): 1.Divide by a on both sides (lead coefficient = 1) 2.Put variables on one side, constants on the other. 3.Complete the square (take ½ of x coefficient and square it – add this number to both sides) 4.Solve by applying the square root property

7.2 Completing the Square Review: x 4 + y 4 – can be factored by completing the square

7.2 Completing the Square Example: Complete the square: Factor the difference of two squares:

7.3 The Quadratic Formula Solving ax 2 + bx + c = 0: Dividing by a: Subtract c/a: Completing the square by adding b 2 /4a 2 :

7.3 The Quadratic Formula Solving ax 2 + bx + c = 0 (continued): Write as a square: Use square root property: Quadratic formula:

7.3 The Quadratic Formula Quadratic Formula: is called the discriminant. If the discriminant is positive, the solutions are real If the discriminant is negative, the solutions are imaginary

7.3 The Quadratic Formula Example:

7.3 The Quadratic Formula Complex Numbers and the Quadratic Formula Solve x 2 – 2x + 2 = 0

7.3 The Quadratic Formula MethodAdvantagesDisadvantages FactoringFastest methodNot always factorable Square root property Not always this form Completing the square Can always be used Requires a lot of steps Quadratic Formula Can always be used Slower than factoring

7.4 The Graph of the Quadratic Function A quadratic function is a function that can be written in the form: f(x) = ax 2 + bx + c The graph of a quadratic function is a parabola. The vertex is the lowest point (or highest point if the parabola is inverted

7.4 The Graph of the Quadratic Function Vertical Shifts: The parabola is shifted upward by k units or downward if k < 0. The vertex is (0, k) Horizontal shifts: The parabola is shifted h units to the right if h > 0 or to the left if h < 0. The vertex is at (h, 0)

7.4 The Graph of the Quadratic Function Horizontal and Vertical shifts: The parabola is shifted upward by k units or downward if k 0 or to the left if h < 0 The vertex is (h, k)

7.4 The Graph of the Quadratic Function Graphing: 1.The vertex is (h, k). 2.If a > 0, the parabola opens upward. If a < 0, the parabola opens downward (flipped). 3.The graph is wider (flattened) if The graph is narrower (stretched) if

7.4 The Graph of the Quadratic Function Vertex = (h, k)

7.4 The Graph of the Quadratic Function Vertex Formula: The graph of f(x) = ax 2 + bx + c has vertex

7.4 The Graph of the Quadratic Function Graphing a Quadratic Function: 1.Find the y-intercept (evaluate f(0)) 2.Find the x-intercepts (by solving f(x) = 0) 3.Find the vertex (by using the formula or by completing the square) 4.Complete the graph (plot additional points as needed)

18.1 Ratio and Proportion Ratio – quotient of two quantities with the same units Note: Sometimes the units can be converted to be the same.

18.1 Ratio and Proportion Proportion – statement that two ratios are equal: Solve using cross multiplication:

18.1 Ratio and Proportion Solve for x: Solution:

18.1 Ratio and Proportion Example: E(volts)=I(amperes)  R(ohms) How much current for a circuit with 36mV and resistance of 10 ohms?

18.2 Variation Types of variation: 1.y varies directly as x: 2.y varies inversely as x: 3.y varies directly as the square of x: 4.y varies directly as the square root of x:

18.2 Variation Solving a variation problem: 1.Write the variation equation. 2.Substitute the initial values and solve for k. 3.Rewrite the variation equation with the value of k from step 2. 4.Solve the problem using this equation.

18.2 Variation Example: If t varies inversely as s and t = 3 when s = 5, find s when t = 5 1.Give the equation: 2.Solve for k: 3.Plug in k = 15: 4.When t = 5:

B.1 Introduction to the Metric System Metric system base units:

B.1 Introduction to the Metric System Multiple in decimal form Power of 10PrefixSymbol megaM kilok hectoh dekada base unit decid centic millim micro 

B.1 Introduction to the Metric System 1 gram = weight of 1 ml of water Unit of weight = 1Kg = 1000 grams 1 liter of water weighs 1 Kg

B.2 Reductions and Conversions Conversions: Note: In Canada, speed is in kph instead of mph

B.2 Reductions and Conversions Conversions

B.2 Reductions and Conversions Try these: –3.5 liters = ________ ml – liter = ________ cc