Lesson 27 Connecting the parabola with the quadratic function

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

Parent function 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 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 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 Find the zeros and vertex of y= -x 2 +3x-2 1) graph y=-x 2 +3x-2 2)Press 2nd 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 x- coordinate 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 2nd Graph, can also be used by locating the zeros in the 2nd column

Vertex & axis of symmetry The x coordinate of the vertex of a parabola is x = -b/2a. 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/2a

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

graph f(x) = x 2 +6x+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