Ch3/4 Lesson 4 Introduction to Domain and Range Solving QF by Graphing © Copyright all rights reserved to Homework depot: www.BCMath.ca

Review) Terms: Domain and Range Domain: All possible x-values (Input Numbers) that are allowed in a function Range: All possible y-values (Output Numbers) that are allowed in a function Ex: Find the Domain and Range: x y -4 -3 -2 -1 1 2 3 4 5 6 7 8 9

Find the Domain and Range for each Graph: x y -5 -4 -3 -2 -1 1 2 3 4 5 6 8 10 x y -5 -4 -3 -2 -1 1 2 3 4 5 6 8 10 The input number can be any value. The domain can be ‘All Real Number’ The input number begins at -4 and ends at 5. All output (y) values have to be greater or equal to two The output values begin at -2 and goes up to10

Likewise, the output numbers are also integers from 0 to 8. x y 1 2 3 4 5 6 7 x y 1 2 3 4 5 6 7 The input numbers are integers from 0 to 8. There are no inputs between the integers Likewise, the output numbers are also integers from 0 to 8.

x y 1 2 3 4 5 6 7 x y 1 2 3 4 5 6 7

I) What does it mean to solve graphically? Solving graphically means finding the points of intersection between two equations When one side of the equation is zero, that means you are looking for points on the graph where the y-coordinate is zero This side of the equation is a parabola This side of the equation is zero, that means the y-coordinate is zero Overall, you are finding all the x-intercepts (aka: roots) of the parabola

A Quadratic function can have either 2 distinct roots (2 different x intercepts) 2 equal roots (1 distinct root OR double roots) The vertex is on the X-axis No real roots (no x intercepts) The entire parabola is above or below the X-axis x y -2 -1 1 2 x - x y -1 1 2 3 4 x 4 x y -2 -1 1 2 4 x

II) How to Find the roots: There are several ways to find the roots of a quadratic equation [x-intercept] Factor the equation and solve [if you can factor it] Complete the square and solve [section 4.3] Quadratic Formula [section 4.4] Graph and find the roots [ti-83] Press 2nd Trace 2:Zero Enter Left Bound: Move cursor to the left side and press enter Right Bound: Move cursor to the right side and press enter Guess: Press enter © Copyright all rights reserved to Homework depot: www.BCMath.ca

Practice: Find the coordinates of the vertex by graphing With a standard grid, part of the graph is hidden. We need to zoom out to see the entire graph Press 2nd Trace Press Enter Press Enter 2:Zero Enter Press Enter Now look for the coordinates of the X-intercepts Repeat the same process to find the other X-intercept Left X-intercept is (6,0) Right X-intercept is (10,0)

Practice: Graph each equation & find the roots This graph has no real roots because the parabola is above the x-axis This graph has only one real root: x = 3 This graph has only one real root: x = 3 This graph has no real roots because the parabola is above the x-axis There are 2 intercepts at x= -2 and x =4/3 There are 2 intercepts at x= -1 and x = 3

Ex: Find two consecutive even integers with a product of 528. Let the first number be x, and the second number be x + 2. x y -30 -20 -10 10 20 30 -500 -400 -300 -200 -100 The two consecutive even integers are –22, –24 and 22, 24

Ex: The length of a rectangle is 5 cm longer than twice its width Ex: The length of a rectangle is 5 cm longer than twice its width. Its area is 133 cm2. Find its dimensions. The dimensions are 7 cm by 19 cm. x y -10 -5 5 10 -150 -100 -50

III) Finding the Equation Given the Roots When given the roots, you can find the equation by going backwards Ex: Given that the roots of a parabola are 1.5 and – 7 , find the equation: Start with the solution Work backwards Now you have two binomials. Multiply them to get the quadratic equation Make sure you have a “y” variable and not a zero for your quadratic equation

Practice: Given that the roots of a quadratic equation are: 3 and -3 Practice: Given that the roots of a quadratic equation are: 3 and -3.5, which of the following is the correct equation?