Chapter 6.2 Solving Quadratic Functions by Graphing Standard & Honors Algebra II Mr. Gilbert Chapter 6.2 Solving Quadratic Functions by Graphing Standard & Honors 11/14/2018

Agenda Warm up Homework Check your answers Applet Lesson 11/14/2018

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Solving Quadratic Functions by Graphing Example 1 Two Real Solutions (4) Example 2 One Real Solution (3) Example 3 No Real Solution (4) Example 4 Estimate Roots (3) Example 5 Write and Solve an Equation (5) 11/14/2018 Lesson 2 Contents

x –1 1 2 3 4 f (x) –4 –6 Solve by graphing. Graph the related quadratic function The equation of the axis of symmetry is Make a table using x values around Then graph each point. x –1 1 2 3 4 f (x) –4 –6 11/14/2018 Example 2-1a

Answer: The solutions of the equation are –1 and 4. From the table and the graph, we can see that the zeroes of the function are –1 and 4. Answer: The solutions of the equation are –1 and 4. 11/14/2018 Example 2-1a

Check Check the solutions by substituting each solution into the original equation to see if it is satisfied. 11/14/2018 Example 2-1a

Solve by graphing. Answer: –3 and 1 11/14/2018 Example 2-1b

Write the equation in form. Solve by graphing. Write the equation in form. Add 4 to each side. Graph the related quadratic function 11/14/2018 Example 2-2a

Notice that the graph has only one x-intercept, 2. Answer: The equation’s only solution is 2. 11/14/2018 Example 2-2a

Solve by graphing. Answer: 3 11/14/2018 Example 2-2b

Explore Let one of the numbers. Then Number Theory Find two real numbers whose sum is 4 and whose product is 5 or show that no such numbers exist. Explore Let one of the numbers. Then Plan Since the product of the two numbers is 5, you know that Original equation Distributive Property Add x2 and subtract 4x from each side. 11/14/2018 Example 2-3a

Solve You can solve by graphing the related function . 11/14/2018 Example 2-3a

Examine Try finding the product of several numbers whose sum is 4. Notice that the graph has no x-intercepts. This means that the original equation has no real solution. Answer: It is not possible for two numbers to have a sum of 4 and a product of 5. Examine Try finding the product of several numbers whose sum is 4. 11/14/2018 Example 2-3a

Answer: no such numbers exist Number Theory Find two real numbers whose sum is 7 and whose product is 14 or show that no such numbers exist. Answer: no such numbers exist 11/14/2018 Example 2-3b

The equation of the axis of symmetry of the related function is Solve by graphing. If exact roots cannot be found, state the consecutive integers between which the roots are located. The equation of the axis of symmetry of the related function is 11/14/2018 Example 2-4a

The x-intercepts of the graph are between 0 and 1 and between 5 and 6. Answer: One solution is between 0 and 1 and the other is between 5 and 6. 11/14/2018 Example 2-4a

Answer: between 0 and 1 and between 3 and 4 Solve by graphing. If exact roots cannot be found, state the consecutive integers between which the roots are located. Answer: between 0 and 1 and between 3 and 4 11/14/2018 Example 2-4b

We need to find t when and Royal Gorge Bridge The highest bridge in the United States is the Royal Gorge Bridge in Colorado. The deck of the bridge is 1053 feet above the river below. Suppose a marble is dropped over the railing from a height of 3 feet above the bridge deck. How long will it take the marble to reach the surface of the water, assuming there is no air resistance? Use the formula where t is the time in seconds and h0 is the initial height above the water in feet. We need to find t when and Original equation 11/14/2018 Example 2-5a

Then locate a right bound and press twice. Graph the related function using a graphing calculator. Adjust your window so that the x-intercepts are visible. Use the zero feature, [CALC], to find the positive zero of the function, since time cannot be negative. Use the arrow keys to locate a left bound for the zero and press . 2nd ENTER Then locate a right bound and press twice. ENTER 11/14/2018 Example 2-5a

Answer:. The positive zero of the function is approximately 8 Answer: The positive zero of the function is approximately 8. It should take about 8 seconds for the marble to reach the surface of the water. 11/14/2018 Example 2-5a

Hoover Dam One of the largest dams in the United States is the Hoover Dam on the Colorado River, which was built during the Great Depression. The dam is 726.4 feet tall. Suppose a marble is dropped over the railing from a height of 6 feet above the top of the dam. How long will it take the marble to reach the surface of the water, assuming there is no air resistance? Use the formula where t is the time in seconds and h0 is the initial height above the water in feet. 11/14/2018 Example 2-5b

Answer: about 7 seconds 11/14/2018 Example 2-5b

Homework Review 11/14/2018

Homework - Honors See Syllabus 6.2 pp. 297-298: 15-36 (multiples of 3), 38-47 11/14/2018

Homework See Syllabus 6.2 pp. 297 – 299 14-19,20-36 even,38-46 11/14/2018