Chapter 5 Lesson 1 Graphing Quadratic Functions

Vocabulary Quadratic Function- A function described by f(x)=ax 2 +bx+c where a≠0 Quadratic Term- ax 2 from the quadratic function Linear Term- bx from the quadratic function Constant Term- c from the quadratic function Parabola- The graph of any quadratic function Axis of Symmetry- A line about which a figure is symmetric Vertex- The point at which the axis of symmetry intersects the parabola Minimum Value- The y-coordinate of the vertex of the quadratic equation where a>0 Maximum Value The y-coordinate of the vertex of the quadratic equation where a<0

Finding the Y-Intercept Given the equation f(x)= ax 2 +bx+c The y-intercept is c Example F(x)= 2x 2 + 4x + 7 Y-intercept is 7 F(x)= -x 2 – 3x -5 Y-intercept is -5

Examples x 2 + 4x – 5 2x 2 + 3x x 2 + 2x – 10

Finding the Axis of Symmetry Given the equation f(x)= ax 2 + bx + c The axis of symmetry is at x = –b / 2a Example F(x) = 3x 2 + 4x +9 Axis of symmetry is x = -4 / 2(3) = -4 / 6 = -2 / 3, x = -2 / 3 F(x) = -2x 2 – 3x + 3 Axis of symmetry is x = -(-3) / 2(-2) = 3 / -4 = - 3 / 4, x = - 3 / 4

Finding the Vertex The x-coordinate of the vertex = the axis of symmetry Example F(x) = 3x 2 + 4x +9 X-coordinate of vertex is x = -4 / 2(3) = -4 / 6 = -2 / 3, x = -2 / 3 F(x) = -2x 2 – 3x + 3 X-coordinate of vertex is x = -(-3) / 2(-2) = 3 / -4 = - 3 / 4, x = - 3 / 4

Examples x 2 + 4x – 3 -2x 2 - 6x + 4 5x 2 + 6x – 2 -x 2 + x – 1

Make A Table of Values that Includes the Vertex To sketch a graph of the parabola by hand, you need to make a table of values that includes the vertex. This is to be done after you have already found the vertex. It is a good idea to pick 2 or 3 points on each side of the vertex to get a good point.

Example F(x)= x 2 +8x+9 Vertex has an x coordinate of -4 xx 2 +8x+9F(x)(x,f(x)) -6(-6) 2 +8(-6)+9-3(-6,-3) -5(-5) 2 +8(-5)+9-6(-5,-6) -4(-4) 2 +8(-4)+9-7(-4,-7) -3(-3) 2 +8(-3)+9-6(-3,-6) -2(-2) 2 +8(-2)+9-3(-2,-3)

Example F(x) = 3x 2 – 12x + 4 x3x 2 – 12x + 4F(x)(x,f(x))

Determining Whether a Function has a Maximum or a Minimum Given the function f(x)= ax 2 + bx + c If a>0, then the parabola opens up and has a minimum value If a<0, then the parabola opens down and has a maximum value Example f(x)= 3x 2 + 2x + 8 f(x)= -1x x + 4

Finding the Maximum/Minimum To find the x-value, do the same process as finding the x value of a vertex. To find the y-value, plug the value for x into the equation for x and solve for y Example F(x) = 2x 2 + 8x + 3 F(x) = -3x 2 + 6x + 1

Domain and Range For Quadratic Functions Domain is all real numbers Range depends on value of a If vertex is maximum value, range is y ≤ maximum value If vertex is minimum value, range is y ≥ minimum value

Homework Worksheet 5-1