4.1 Graph Quadratic Functions in

Transcription

1 4. Graph Quadratic Functions in Standard Form Goal p Graph quadratic functions. Your Notes VOCABULARY Quadratic function Parabola Verte Ais of smmetr Minimum and maimum value PARENT FUNCTION FOR QUADRATIC FUNCTIONS The parent function for the famil of all quadratic functions is f() 5. The graph is shown below. The ais of smmetr divides the parabola into mirror images and passes through the verte. The lowest or highest point on a parabola is the verte. The verte for f() = is (0, 0). 5 For f() 5 a, and for an quadratic function g()5 a b c where b 5 0, the verte lies on the and the ais of smmetr is Lesson 4. Algebra Notetaking Guide Copright Holt McDougal. All rights reserved.

2 4. Graph Quadratic Functions in Standard Form Goal p Graph quadratic functions. Your Notes VOCABULARY Quadratic function A function that can be written in the standard form 5 a b c where a Þ 0 Parabola The U-shaped graph of a quadratic function Verte The lowest or highest point on a parabola Ais of smmetr The vertical line that divides the parabola into mirror images and passes through the verte Minimum and maimum value For 5 a b c, the verte s -coordinate is the minimum value of the function if a > 0 and its maimum value if a < 0. PARENT FUNCTION FOR QUADRATIC FUNCTIONS The parent function for the famil of all quadratic functions is f() 5. The graph is shown below. The ais of smmetr divides the parabola into mirror images and passes through the verte. The lowest or highest point on a parabola is the verte. The verte for f() = is (0, 0). 5 For f() 5 a, and for an quadratic function g()5 a b c where b 5 0, the verte lies on the -ais and the ais of smmetr is Lesson 4. Algebra Notetaking Guide Copright Holt McDougal. All rights reserved.

3 Choose values of on both sides of the ais of smmetr 5 0. Eample Graph 5. Compare the graph with the graph of 5.. Make a table of values for 5. 0 Graph a function of the form 5 a c. Plot the points from the table. 3. Draw a smooth through 5 the points. 4. Compare the graphs of 5 and 5. Both graphs have the same. However, the graph of 5 opens and is than the graph of 5. Also, its verte is units higher. PROPERTIES OF THE GRAPH OF 5 a b c Characteristics of the graph of 5 a b c: The graph opens up if a 0 and opens down if a 0. The graph is narrower than the graph of 5 if a and wider if a. The ais of smmetr is 5 has -coordinate. and the verte The -intercept is. So, the point (0, ) is on the parabola. Copright Holt McDougal. All rights reserved. Lesson 4. Algebra Notetaking Guide 89

4 Choose values of on both sides of the ais of smmetr 5 0. Eample Graph a function of the form 5 a c Graph 5. Compare the graph with the graph of 5.. Make a table of values for Plot the points from the table. 3. Draw a smooth curve through the points. 4. Compare the graphs of 5 and 5. Both graphs have the same ais of smmetr. However, the graph of 5 opens 5 (0, ) (, 0) (, 0) 5 (, 6) (, 6) down and is narrower than the graph of 5. Also, its verte is units higher. PROPERTIES OF THE GRAPH OF 5 a b c Characteristics of the graph of 5 a b c: The graph opens up if a > 0 and opens down if a < 0. The graph is narrower than the graph of 5 if a > and wider if a <. b The ais of smmetr is 5 } and the verte a b has -coordinate } a. The -intercept is c. So, the point (0, c ) is on the parabola. Copright Holt McDougal. All rights reserved. Lesson 4. Algebra Notetaking Guide 89

5 Be sure to include the negative sign before the fraction when calculating the coordinate of the verte. Unless stated otherwise, the domain of a quadratic function is all real numbers. Depending on the sign of a, the range will either be all real numbers greater than or equal to the minimum value or all real numbers less than or equal to the maimum value. Eample Graph Graph a function of the form 5 a b c. Identif the coefficients of the function. The coefficients are a 5, b 5, and c 5. Because a 0, the parabola opens.. Find the verte. First, calculate the -coordinate. 5 b } a 5 5 Then find the -coordinate. 5 5 The verte is (, ). Plot this point. 3. Draw the ais of smmetr Identif the -intercept c, which is. Plot the point (0, ). Then reflect this point in the ais of smmetr to plot another point (4, ). 5. Evaluate the function for another value of, such as Plot the point (, ) and its reflection (3, ) in the ais of smmetr. 6. Draw a parabola through the plotted points. MINIMUM AND MAXIMUM VALUES Words For 5 a b c, the verte's -coordinate is the minimum value of the function if a 0 and the maimum value if a 0. Graphs minimum a is positive maimum a is negative. 90 Lesson 4. Algebra Notetaking Guide Copright Holt McDougal. All rights reserved.

6 Be sure to include the negative sign before the fraction when calculating the coordinate of the verte. Unless stated otherwise, the domain of a quadratic function is all real numbers. Depending on the sign of a, the range will either be all real numbers greater than or equal to the minimum value or all real numbers less than or equal to the maimum value. Eample Graph Identif the coefficients of the function. The coefficients are a 5, b 5 4, and c 5 3. Because a < 0, the parabola opens down.. Find the verte. First, calculate the (, ) -coordinate. (3, 0) 5 } b a 5 4 } () Graph a function of the form 5 a b c 5 Then find the -coordinate. 5 () 4() 3 5 (, 0) (0, 3) (4, 3) 5 The verte is (, ). Plot this point. 3. Draw the ais of smmetr Identif the -intercept c, which is 3. Plot the point (0, 3 ). Then reflect this point in the ais of smmetr to plot another point (4, 3 ). 5. Evaluate the function for another value of, such as 5. 5 () 4() Plot the point (, 0 ) and its reflection (3, 0 ) in the ais of smmetr. 6. Draw a parabola through the plotted points. MINIMUM AND MAXIMUM VALUES Words For 5 a b c, the verte's -coordinate is the minimum value of the function if a > 0 and the maimum value if a < 0. Graphs minimum a is positive maimum a is negative. 90 Lesson 4. Algebra Notetaking Guide Copright Holt McDougal. All rights reserved.

7 Eample 3 Find the minimum or maimum value Tell whether the function has a minimum value or a maimum value. Then find it and state the domain and range. Because a 0, the function has a value. To find it, calculate the coordinates of the verte. 5 b } a The maimum value is 5. The domain is all numbers. The range is all numbers than or equal to. Checkpoint Complete the following eercises. Graph the function } Graph the function. Label the verte and ais of smmetr. 5 4 Homework 4. Find the minimum value and range of Copright Holt McDougal. All rights reserved. Lesson 4. Algebra Notetaking Guide 9

8 Eample 3 Find the minimum or maimum value Tell whether the function has a minimum value or a maimum value. Then find it and state the domain and range. Because a < 0, the function has a maimum value. To find it, calculate the coordinates of the verte. 5 } b a 5 } 5 (3) 5 3() () The maimum value is 5 6. The domain is all real numbers. The range is all real numbers less than or equal to 6. Checkpoint Complete the following eercises. Graph the function } Graph the function. Label the verte and ais of smmetr Homework (, ) Find the minimum value and range of ; all real numbers greater than or equal to.5 Copright Holt McDougal. All rights reserved. Lesson 4. Algebra Notetaking Guide 9