Quadratics functions: review

Transcription

1 December 18, 2008 Quadratic functions: review page 1 Quadratics functions: review Monday, December 15, we will have a quiz on quadratic functions. Today s packet has review problems and some additional applications. Review outline Quadratic function general form: f(x) = ax 2 + bx + c Quadratic function tables and graphs (parabolas) Important places on the parabola graph [see chart below] vertex (minimum or maximum) axis of symmetry zeros (x-intercepts) y-intercept how to find on graph how to find in table how to find by calculation on calculator lowest or highest point vertical line through vertex where graph crosses x-axis where graph crosses y-axis point where output values change direction x =! b, then use f(x) formula to get the y-coordinate 2 nd CALC min. or max. find vertex (see above)! b find vertex x = (see above) points where y = 0 (will study later) 2 nd CALC zero point where x = 0 (0, c) use x = 0 row in table Direction of the parabola given by the sign of a ( when a > 0, when a < 0) Using symmetry to find missing points in a graph or table Using symmetry of parabolas x-coordinate of the vertex is the average of the zeros symmetry allows you fill in missing points in a graph or table Quadratic functions in real-world applications Review problems 1. The graph of a quadratic function f(x) is shown. Answer these questions about it. a. What are the zero(s)? b. What is the y-intercept? c. What is the vertex? d. What is the equation for the axis of symmetry? e. In the function formula f(x) = ax 2 + bx + c, is the value of a positive or negative? f. In the function formula f(x) = ax 2 + bx + c, what must be the value of c?

2 December 18, 2008 Quadratic functions: review page 2 2. Here is a quadratic function: f(x) = 2x 2 8x 8. a. Make a graph and a table. Use your calculator as little as possible. x f(x) b. What are the coordinates of the vertex? c. What are the coordinates of the zero(s)? d. What are the solutions to the equation 2x 2 8x 8 = 0? 3. Answer these questions about quadratic function f(x) = 2x x 7. a. Make a small sketch of the shape of the graph of f(x). b b. Find (x, y) coordinates of the vertex. Hint: x =!. c. Is the vertex a maximum point or a minimum point? Hint: Think about the shape from part a. d. Find the (x, y) coordinates of the y-intercept.

3 December 18, 2008 Quadratic functions: review page 3 4. Solve these equations graphically on your calculator. Remember the first step is to rewrite the equation so that it has a 0 on one side. a. x = x 2 20 Solutions: b. 21 x 2 = 2x 2 Solutions: 5. For the function f(x) = x 2 + 2x + 11, do all of the following on your calculator. Note: The reason this is a good calculator problem is that most of the answers aren t whole numbers. a. Make a table and a graph. Be sure that your graph includes all of the important points we have discussed. If any of them are off the screen, press [WINDOW] and adjust the settings. Your [WINDOW] settings Xmin = Xmax = Ymin = Ymax = b. Find the coordinates of the x-intercepts. c. Find the coordinates of the y-intercept. d. Find the coordinates of the vertex. e. Write an equation for the axis of symmetry.

4 December 18, 2008 Quadratic functions: review page 4 6. This is a calculator-based problem. You should use the calculator methods you ve learned. Be sure to tell what you did on the calculator. A golf ball is hit into the air. Its height is given by the equation y = 16x x where x stands for the time in seconds and y stands for the height in feet. a. On your calculator, make a table and a graph. Be sure that your graph includes all of the important points we have discussed. Adjust the window if necessary. Record what you do and see on the calculator. Your WINDOW settings Xmin = Xmax = Ymin = Ymax = Table Graph b. At what x-value(s) is the ball on the ground? (Show or tell how you get your answer.) c. At what time is the golf ball at its greatest height? (Show or tell how you get your answer.) d. What is the highest height that the golf ball reaches? (Show or tell how you get your answer.)

5 December 18, 2008 Quadratic functions: review page 5 7. Each row in this table describes a parabola. Fill in the blanks with the missing information.! b Hint: x =. function formula axis of symmetry vertex a. f(x) = 3x 2 12x 10 x = (, ) b. f(x) = x x + 1 x = 3 (, ) c. f(x) = 2x 2 + x + 30 x = ( 4, ) 8. Here is a table of values for a quadratic function f(x), with a few numbers missing. a. What point is the vertex, and is it a minimum or a maximum? vertex: (, ) circle which: maximum b. Fill in the missing numbers in the table. c. What are the zeros of this function? minimum x f(x) d. It s a fact that for any quadratic with two zeros, the x-coordinate of the vertex is the average of the x-coordinates of zeros. Do the arithmetic that confirms this is true in this problem.