Section 9.3 Graphing Quadratic Functions

Transcription

1 Section 9.3 Graphing Quadratic Functions A Quadratic Function is an equation that can be written in the following Standard Form., where a 0. Every quadratic function has a U-shaped graph called a. If the leading coefficient, ( ), is, the parabola. If the leading coefficient is, the parabola in the shape of an upside down U. Example A: Complete each table for the given functions then graph the points. y = x 2 x y y = x 2 x y Solutions: 2 2

2 y = x 2 y = x 2 a =1 a = 1 Opens upward Opens downward The is the lowest point of a parabola that opens up and the highest point of a parabola that opens down. A line that passes through the vertex that divides the parabola into two symmetric parts is called the. The symmetric parts are mirror images of each other. In the examples above, x = 0 is the axis of symmetry. The axis of symmetry will always be a vertical line. Graph of a Quadratic Function:

3 Steps to Graphing a Quadratic Function: 1) Find the x-coordinate of the vertex. 2) Make a table of values, using x-values to the left and right of the vertex. 3) Plot the points and connect them with a smooth curve to form a parabola. Example B: Complete the following steps for the function y = x 2 +6x + 8 a) Tell whether the graph of the function opens up or down. b) Find the coordinates of the vertex. c) Write the equation of the axis of symmetry. a) a =. b) Evaluate b 2a when a = ( ) and b = ( ). x = b 2a, So the coordinates of (, ). c) The equations of the line of symmetry is x =( ).

4 Practice: Complete the following steps for each function. a) Tell whether the graph of the function opens up or down. b) Find the coordinates of the vertex. c) Write the equation of the axis of symmetry. 1) y = 2x 2 8x + 3 2) y = x 2 + x 2 3) y = 2x 2 + 5x 4 a) opens a) opens a) opens b) (, ) b) (, ) b) (, ) c) c) c)

5 Example C: Graph the equations y = x 2 2x 3. Before you begin, identify a, b, and c. a = 1, b = -2, and c = -3 1) Find the x-coordinate of the vertex. b 2a when a = ( ) and b = ( ) Substitute and simplify. 2) Make a table of values, using x-values to the left and right of x =. Axix of symmetry x y 1 3) Plot the points. The vertex is (, ) and the axis of symmetry is x =. Connect the points to form a parabola that opens up since a is positive.

6 Example D: Graph the equations y = 2x 2 x + 2. a = -2, b = -1, and c = 2 1) Find the x-coordinate of the vertex. b 2a when a = ( ) and b = ( ) Substitute and simplify. 2) Make a table of values, using x-values left and right of x =. x 1 4 y 3) Plot the points. The vertex is (, ) and the axis of symmetry is x = ( ). Connect the points to form a parabola that opens down since a is negative.

7 Note: At the vertex, there is either a maximum or minimum y-value. If the parabola opens up, then there is a. If the parabola opens down, then there is a. Example E: Determine the maximum or minimum for the function y = x 2 + 4x + 7. First, the parabola opens up since a, so there will be a value of y. Next, find the x-coordinate of the vertex. b 2a when a = ( ) and b = ( ) x = Substitute the x-value into the function and solve for y. y = x 2 + 4x + 7 So, the minimum y-value is y =.

8 Example F: If an object is dropped from a height of 80 feet, the function d = 16t gives the height of the object after t seconds. Graph this function. Approximately how long does it take the object to reach the ground (d = 0)? First, make a table of values. Since t = time in seconds, you will not use negative values. t d Graph the values: Since the height goes from positive to negative between seconds, the object passes zero feet in height and there for reaches the ground in approximately seconds.

9 Example G: You throw a basketball whose path can be modeled by the equation y = 16x 2 +15x + 6, where x represents time (in seconds) and y represents height of the basketball (in feet). What is the maximum height that the basketball reaches? First, find the x-coordinate of the vertex. b 2a when a = -16 and b = 15 x = So the ball will be at its maximum height at.47 seconds after it leaves your hand. Substitute the x-value into the function and solve for y. So, the maximum y-value is which means the ball reach a maximum height of approximately.