Warm-Up Exercises Find the zeros of the polynomial function. 1.f(x) = x 2 + 5x – 36 2.f(x) = x 2 – 9x + 20 ANSWER –9, 4 ANSWER 4, 5

Warm-Up Exercises ANSWER 9 in. Find the zeros of the polynomial function. You are making a rectangular flag for your school team. The number of square inches in the area of the flag is 9x The number of inches in the length of the flag is x What is the width of the flag? 3.

Warm-Up Exercises EXAMPLE 1 Solve a quadratic equation having two solutions Solve x 2 – 2x = 3 by graphing. STEP 1 Write the equation in standard form. Write original equation. x 2 – 2x = 3 Subtract 3 from each side. x 2 – 2x – 3 = 0 STEP 2 Graph the function y = x 2 – 2x – 3. The x- intercepts are – 1 and 3. SOLUTION

Warm-Up Exercises EXAMPLE 1 ANSWER The solutions of the equation x 2 – 2x = 3 are – 1 and 3. Solve a quadratic equation having two solutions You can check – 1 and 3 in the original equation. x 2 – 2x = 3 (–1) 2 –2(–1) 3 = ? (3) 2 –2(3) 3 = ? 3 = 3 Write original equation. Substitute for x. Simplify. Each solution checks. CHECK

Warm-Up Exercises EXAMPLE 2 Solve a quadratic equation having one solution Solve – x 2 + 2x = 1 by graphing. SOLUTION STEP 1 Write the equation in standard form. Write original equation. – x 2 + 2x = 1 Subtract 1 from each side. – x 2 + 2x – 1 = 0

Warm-Up Exercises EXAMPLE 2 Solve a quadratic equation having one solution STEP 2 Graph the function y = – x 2 + 2x – 1. The x- intercept is 1. ANSWERThe solution of the equation – x 2 + 2x = 1 is 1.

Warm-Up Exercises EXAMPLE 3 Solve a quadratic equation having no solution Solve x = 4x by graphing. SOLUTION STEP 1 Write the equation in standard form. Write original equation. x = 4x Subtract 4x from each side. x 2 – 4x + 7 = 0

Warm-Up Exercises EXAMPLE 3 Solve a quadratic equation having no solution STEP 2 Graph the function y = x 2 – 4x + 7. The graph has no x -intercepts. ANSWER The equation x = 4x has no solution.

Warm-Up Exercises GUIDED PRACTICE for Examples 1, 2 and 3 Solve the equation by graphing. 1. x 2 – 6x + 8 = 0 ANSWER 2, 4 2. x 2 + x = – 1 ANSWER no solution 3. – x 2 + 6x = 9 ANSWER 3

Warm-Up Exercises EXAMPLE 4 Find the zeros of a quadratic function Find the zeros of f(x) = x 2 + 6x – 7. SOLUTION Graph the function f(x) = x 2 + 6x –7. The x- intercepts are – 7 and 1. ANSWER The zeros of the function are – 7 and 1. CHECK Substitute – 7 and 1 in the original function. f(– 7) = (– 7) 2 + 6(– 7) – 7 = 0 f(1) = (1) 2 + 6(1) – 7 = 0

Warm-Up Exercises EXAMPLE 5 Approximate the zeros of a quadratic function Approximate the zeros of f(x) = x 2 + 4x + 1 to the nearest tenth. Graph the function f(x) = x 2 + 4x + 1. There are two x- intercepts: one between –4 and –3 and another between –1 and 0. STEP 1 SOLUTION

Warm-Up Exercises EXAMPLE 5 STEP 2 Make a table of values for x- values between – 4 and – 3 and between – 1 and 0 using an increment of 0.1. Look for a change in the signs of the function values. x–3.9– 3.8–3.7–3.6–3.5–3.4–3.3–3.2–3.1 f(x) –0.11–0.44–0.75–1.04–1.31–1.56–1.79 Approximate the zeros of a quadratic function

Warm-Up Exercises EXAMPLE 5 Approximate the zeros of a quadratic function x – 0.9– 0.8– 0.7– 0.6– 0.5– 0.4– 0.3– 0.2– 0.1 f(x) – 1.79– 1.56– 1.31–1.04– 0.75– 0.44– ANSWER In each table, the function value closest to 0 is – So, the zeros of f(x) = x 2 + 4x + 1 are about – 3.7 and about – 0.3.

Warm-Up Exercises GUIDED PRACTICE for Examples 4 and 5 4. Find the zeros of f(x) = x 2 + x – 6. ANSWER The zeros of the function are – 3 and Approximate the zeros of f(x) = –x 2 + 2x + 2 to the nearest tenth. ANSWER The zeros of the function are –0.7 and 2.7.

Warm-Up Exercises EXAMPLE 6 Solve a multi-step problem An athlete throws a shot put with an initial vertical velocity of 40 feet per second as shown. a. Write an equation that models the height h (in feet) of the shot put as a function of the time t (in seconds) after it is thrown. b. Use the equation to find the time that the shot put is in the air. SPORTS

Warm-Up Exercises Solve a multi-step problem EXAMPLE 6 a. Use the initial vertical velocity and the release height to write a vertical motion model. h = – 16t 2 + vt + s Vertical motion model h = – 16t t Substitute 40 for v and 6.5 for s. b. The shot put lands when h = 0. To find the time t when h = 0, solve 0 = – 16t t for t. To solve the equation, graph the related function h = – 16t t on a graphing calculator. Use the trace feature to find the t -intercepts.

Warm-Up Exercises Solve a multi-step problem EXAMPLE 6 ANSWER There is only one positive t -intercept. The shot put is in the air for about 2.6 seconds.

Warm-Up Exercises GUIDED PRACTICE for Example 6 WHAT IF? In Example 6, suppose the initial vertical velocity is 30 feet per second. Find the time that the shot put is in the air. 6. ANSWER About 2.1 seconds

Warm-Up Exercises Daily Homework Quiz 1. Solve x 2 + 6x + 8 by graphing. – 4, – 2 ANSWER

Warm-Up Exercises Daily Homework Quiz 2. x 2 + 6x = – 10 ANSWERnone 3. x 2 + 6x = – 9 oneANSWER Find the number of solutions for each equation.

Warm-Up Exercises Daily Homework Quiz 4. Find the zeros of f(x) = – x 2 + 2x + 3. – 1, 3 ANSWER 5. Approximate the zeros of f(x) = x 2 + x – 3 to the nearest tenth. ANSWER – 2.3, 1.3

Warm-Up Exercises Daily Homework Quiz 6. Maria throws a shot put with an initial velocity of 25 feet per second. She releases it at a height of 5 feet. Find the time the shot put is in the air. about 1.7 sec ANSWER