Lesson 27 Connecting the parabola with the quadratic

Lesson 27 Connecting the parabola with the quadratic function

Quadratic equation and quadratic function n A quadratic equation is one that can be written in the form ax 2+bx+c=0 A quadratic function is a function that can be written in the form f(x) = ax 2+bx+c, which is called the standard form of a quadratic function, where a is not 0, and a, b, c are real numbers. Because f(x) = y, a quadratic function can also be written as y = ax 2+bx+c

Converting to standard form n n n n Write y-6=2 x-x 2 in standard form Isolate y and list the terms in decreasing order. y= -x 2+2 x+6 Convert f(x) =2(x-4)2+9 into standard form = 2(x 2 -8 x+16)+9 = 2 x 2 -16 x+32+9 = 2 x 2 -16 x+41

Parent function n n The parent function of a quadratic equation is f(x) = x 2, so in standard form a= 1, b= 0, c= 0

Graph of a quadratic function n The graph is called a parabola. The graph of every quadratic function is a parabola. All parabolas have the same symmetric U shape. For the graph of f(x) = x 2, the point (0, 0) is the vertex of the parabola. The vertex indicates where the curve changes direction. It is the lowest(orhighest) point on a parabola

Zeros n n n The zeros of a quadratic function are the values of x for which the function equals 0. On a graph, the zeros are the x-intercepts, or where the graph intersects the x-axis. The y-intercept is the point where a graph intersects the y-axis and can be found by substituting x with 0.

Finding the zeros and vertex of a parabola n n n n Find the zeros and vertex of y= -x 2+3 x-2 1) graph y=-x 2+3 x-2 2)Press 2 nd Trace to access the CALC menu. Select 2: Zero. Choose a point to the left and right of the x-intercepts and press enter. It will show you the root (x-intercept). Repeat this for all roots. 3) To find the y-intercept, press TRACE and enter 0, after X= 4) Since the vertex is halfway between the x-intercepts, the xcoordinate of the vertex is the average of the 2 x-coordinates. With TRACE still chosen, enter the average you just found for x= Now you have the coordinates of the vertex 5) The TABLE menu , which can be accessed from 2 nd Graph, can also be used by locating the zeros in the 2 nd column

Vertex & axis of symmetry n n n The x coordinate of the vertex of a parabola is x = -b/2 a. The y coordinate can be found by substitution. The vertex is on the axis of symmetry of the parabola. The axis of symmetry is a line that divides a figure into 2 congruent mirror images. Therefore, the reflection of each point on the left side of a parabola is located on the right side of the parabola. The equation of the axis of symmetry is x = -b/2 a

Graphing quadratic equations n n n Graph f(x) = x 2 -x-6 Use x= -b/2 a to find the x coordinate of the vertex, then substitute to find the y coordinate Find the axis of symmetry by using x= -b/2 a Plot the vertex and the axis of symmetry Make a table of ordered pairs to complete the graph.

graph n n n f(x) = x 2 +6 x+9 Identify the x- and y-intercepts. Identify the domain and range f(x) = x 2 -x -6 Identify the x- and y-intercepts Identify the domain and range