6.6 analyzing graphs of quadratic functions
β’Download as PPT, PDFβ’
11 likesβ’26,405 views
![Graphing the parabola y = f ( x ) = ax 2 ,[object Object],0 1 4 1 (β1, 1) (0, 0) (1, 1) (2, 4) y x 4 (β2, 4) Axis of symmetry: x = 0 ( y=x 2 is symmetric with respect to the y-axis ) Vertex (0, 0) When a > 0 , the parabola opens upwards and is called concave up. The vertex is called a minimum point. y 2 1 0 β 1 β 2 x](data:image/gif;base64,R0lGODlhAQABAIAAAAAAAP///yH5BAEAAAAALAAAAAABAAEAAAIBRAA7)
Recommended
Recommended
More Related Content
What's hot
What's hot (20)
Similar to 6.6 analyzing graphs of quadratic functions
Similar to 6.6 analyzing graphs of quadratic functions (20)
More from Jessica Garcia
More from Jessica Garcia (20)
Recently uploaded
Recently uploaded (20)
6.6 analyzing graphs of quadratic functions
- 1. 6.6 Analyzing Graphs of Quadratic Functions
- 5. Graphing the parabola y = f ( x ) = ax 2 + k Algebraic Approach: y = β 4 x 2 β 3 Numerical Approach: Graphical Approach: Consider the equation y = β 4 x 2 β 3 . What is a ? a = β 4 x Vertex (0, -3) -16 -4 0 -4 -16 -4 x 2 -19 -7 -3 -7 -19 - 4 x 2 - 3 2 1 0 β 1 β 2 x
- 6. y = β 4 x 2 β 3 . x Vertex (0, -3) y = β 4 x 2 In general the graph of y = ax 2 + k is the graph of y = ax 2 shifted vertically k units. If k > 0, the graph is shifted up. If k < 0, the graph is shifted down. (P. 267) The graph y = β 4 x 2 is shifted down 3 units.
- 7. a = β 4. What effect does the 3 have on the function? y x y = β 4 x 2 y = β 4( x β 3) 2 Consider the equation y = β 4( x β 3) 2 . What is a ? The axis of symmetry is x = 3. Numerical Approach: Axis of symmetry is shifted 3 units to the right and becomes x = 3 -16 -4 0 -4 -16 β 4 x 2 2 1 0 β 1 β 2 x -4 -16 -36 -64 -100 - 4 ( x- 3) 2 -36 3 0
- 16. Standard 9 Write a quadratic function in vertex form Write y = x 2 β 10 x + 22 in vertex form. Then identify the vertex. y = x 2 β 10 x + 22 Write original function. y + ? = ( x 2 β 10 x + ? ) + 22 Prepare to complete the square. y + 25 = ( x 2 β 10 x + 25 ) + 22 y + 25 = ( x β 5) 2 + 22 Write x 2 β 10 x + 25 as a binomial squared. y + 3 = ( x β 5) 2 Write in vertex form. Add β 10 2 2 ( ) = (β5) 2 = 25 to each side. The vertex form of the function is y + 3 = ( x β 5) 2 . The vertex is (5, β 3). ANSWER
- 17. EXAMPLE 1 Write a quadratic function in vertex form Write a quadratic function for the parabola shown. SOLUTION Use vertex form because the vertex is given. y β k = a ( x β h ) 2 Vertex form y = a ( x β 1) 2 β 2 Substitute 1 for h and β2 for k . Use the other given point, ( 3 , 2 ) , to find a . 2 = a ( 3 β 1) 2 β 2 Substitute 3 for x and 2 for y . 2 = 4 a β 2 Simplify coefficient of a . 1 = a Solve for a .
- 18. EXAMPLE 1 Write a quadratic function in vertex form A quadratic function for the parabola is y = ( x β 1) 2 β 2 . ANSWER
- 19. EXAMPLE 1 Graph a quadratic function in vertex form Graph y β 5 = β ( x + 2) 2 . SOLUTION STEP 1 STEP 2 Plot the vertex ( h , k ) = ( β 2, 5) and draw the axis of symmetry x = β 2. 14 Identify the constants a = β , h = β 2, and k = 5. Because a < 0, the parabola opens down. 14
- 20. EXAMPLE 1 Graph a quadratic function in vertex form STEP 3 Evaluate the function for two values of x . Plot the points (0, 4) and (2, 1) and their reflections in the axis of symmetry. STEP 4 Draw a parabola through the plotted points. x = 0 : y = ( 0 + 2) 2 + 5 = 4 14 β x = 2 : y = ( 2 + 2) 2 + 5 = 1 14 β
- 21. GUIDED PRACTICE for Examples 1 and 2 Graph the function. Label the vertex and axis of symmetry. 1. y = ( x + 2) 2 β 3 2. y = β( x + 1) 2 + 5
- 22. GUIDED PRACTICE for Examples 1 and 2 3. f ( x ) = ( x β 3) 2 β 4 12