LEARNING GOALS - LESSON 5.3 – DAY 1 Warm Up Find the x-intercept of the function. f(x) = –3x + 9 LEARNING GOALS - LESSON 5.3 – DAY 1 5.3.1: Use a graph and table to find zeros of a quadratic function. 5.3.2: Find zeros by factoring, ac method, square roots, GCF A _____ of a function is a value of the input x that makes the output f(x) equal zero. (“When you plug in ______ for _____.”) The zeros of a function are the x-intercepts. What does a zero look like in an equation? What does a zero look like on a graph? How many zeros can a parabola have?

Example 1: Finding Zeros by Using a Graph or Table Find the zeros of f(x) = x2 – 6x + 8 by using a graph and table. Method 1 Graph the function f(x) = x2 – 6x + 8 1. The graph opens __________. 2. The axis of symmetry is:__ The vertex is:(_____,_____) The y-intercept is: ______ Method 2 Use a calculator: Enter y = x2 – 6x + 8 into a graphing calculator. Step 1: Put equation in Y= menu. Step 2: Push Zoom and select option 0: Zoom Fit Step 3: Look for possible x values that will have a zero. Step 4: Push 2nd and then TABLE Step 5: Look for x values that give you a 0 in the y column OR Step 6: Push GRAPH Step 7: Push 2nd and then TRACE Step 8: Select option 2: Zero Step 9: Move cursor to the left of a suspected zero; hit enter. Step 10: Move cursor to the right of the suspected zero; hit enter. Step 11: Hit enter one more time for good measure.

B. Find the zeros of g(x) = –x2 – 2x + 3 by using a graph and a table with your calculator. YOU CAN ALSO FIND ZEROS BY USING ALGEBRA. For example, to find the zeros of f(x)= x2 + 2x – 3, you can set the function equal to zero. We do this because: SETTING THE FUNCTION = to _____ WILL GIVE US THE ZEROS or _____- VALUES OF THE FUNCTION ARE WHERE IT CROSSES THE ____-_________ or WHERE THE FUCNTIONS VALUE IS _________. The solution to a quadratic equation of the form ax2 + bx + c = 0 are roots. The _________ of an equation are the values of the variable that make the equation true. Functions have zeros or x-intercepts. Equations have solutions or roots. Reading Math

Example 2: Finding Zeros by Factoring You can find the roots of some quadratic equations by factoring and applying the Zero Product Property. ZERO PRODUCT PROPERTY: Example 2: Finding Zeros by Factoring A. Find the zeros of the function by factoring. f(x) = x2 – 4x – 12 Set the function equal to 0. Factor using _________________: Find factors of ______ that add to ____. Apply the Zero Product Property. Solve each equation. B. Find the zeros of the function by factoring. g(x) = 3x2 + 18x C. Find the zeros of the function by factoring. f(x)= x2 – 5x – 6 D. Find the zeros of the function by factoring.

LEARNING GOALS - LESSON 5.3 - DAY 2 5.3.3: Find roots of a quadratic using special factors. 5.3.4: Write a quadratic function with given zeros. 5.3.5: Find Max/Min height of a ball. Example 3: Find Roots by Using Special Factors A. Find the roots of the equation by factoring. 4x2 = 25 B. Find the roots of the equation by factoring. 18x2 = 48x – 32 C. Find the roots of the equation by factoring. Rewrite in standard form. x2 – 4x = –4 Factor the perfect-square trinomial. Apply the Zero Product Property. Solve each equation. Solve each equation. D. Find the roots of the equation by factoring. 25x2 = 9

Example 4: Using Zeros to Write Function Rules If you know the zeros of a function, you can work backward to write a rule for the function Example 4: Using Zeros to Write Function Rules Write a quadratic function in standard form with zeros 4 and –7. Write the zeros as solutions for two equations. Rewrite each equation so that it equals 0. Apply the converse of the Zero Product Property to write a product that equals 0. Multiply the binomials. Replace 0 with f(x). Write a quadratic function in standard form with zeros 5 and –5. NOTE: there are many quadratic functions with the same zeros. For example, the functions: A. f(x) = x2 – x – 2 B. g(x) = –x2 + x + 2 C. h(x) = 2x2 – 2x – 4 So writing function rules using zeros does not give a specific function A B C all have zeros at 2 and –1

Example 6: Sports Application Any object that is thrown or launched into the air is a __________ . The general function that approximates the height h in feet of a projectile on Earth after t seconds is given. (The formula below does not account for ________________ or _________________. h(t) = –16t2 + v0t + h0 Example 6: Sports Application A golf ball is hit from ground level with an initial vertical velocity of 80 ft/s. After how many seconds will the ball hit the ground? h(t) = –16t2 + v0t + h0 Write the general projectile function. Substitute ______ for v0 and ______ for h0. The ball will hit the ground when its height is ________. Set h(t) equal to ______. Factor: The GCF is ______t. Apply the Zero Product Property. Solve each equation. B. A football is kicked from ground level with an initial vertical velocity of 48 ft/s. How long is the ball in the air? h(t) = –16t2 + v0t + h0 The ball will hit the ground when its height is __________