Graphing Quadratic Functions Definitions Rules & Examples Practice Problems

Definitions  A quadratic function is described by an equation of the following forms:  Standard Form  Vertex Form 2

Definitions  Graphs of a quadratic function  Always in the shape of a parabola 3

Definitions  Vertex  Minimum or Maximum of the function Minimum is the least value possible of the function Maximum is the greatest value possible of the function  Also intercepts the axis of symmetry  Axis of Symmetry  Vertical line which goes through the exact center of the parabola  Y-intercept  The point where the parabola “intercepts” the y-axis 4

Minima & Maxima  Parabola opens up (minimum) when “a” is positive  Parabola opens down (maximum) when “a” is negative 5

An easy way to remember Min/Max Positive People 6 Negative People Minimum Maximum SmileFrown

Find the line of symmetry of y = 3x 2 – 18x + 7 Finding the Line of Symmetry When a quadratic function is in standard form The equation of the line of symmetry is y = ax 2 + bx + c, For example… Using the formula… Thus, the line of symmetry is x = 3.

Finding the Vertex We know the line of symmetry always goes through the vertex. Thus, the line of symmetry gives us the x – coordinate of the vertex. To find the y – coordinate of the vertex, we need to plug the x – value into the original equation. STEP 1: Find the line of symmetry STEP 2: Plug the x – value into the original equation to find the y value. y = –2x 2 + 8x –3 y = –2(2) 2 + 8(2) –3 y = –2(4)+ 8(2) –3 y = –8+ 16 –3 y = 5 Therefore, the vertex is (2, 5) Note: The y-coordinate of the vertex is also the Minimum/Maximum VALUE of the function

A Quadratic Function in Standard Form The standard form of a quadratic function is given by y = ax 2 + bx + c There are 3 steps to graphing a parabola in standard form. STEP 1: Find the line of symmetry STEP 2: Find the vertex STEP 3: Find two other points and reflect them across the line of symmetry. Then connect the five points with a smooth curve. Plug in the line of symmetry (x – value) to obtain the y – value of the vertex. MAKE A TABLE using x – values close to the line of symmetry. USE the equation

STEP 1: Find the line of symmetry Let's Graph ONE! Try … y = 2x 2 – 4x – 1 A Quadratic Function in Standard Form Thus the line of symmetry is x = 1

Let's Graph ONE! Try … y = 2x 2 – 4x – 1 STEP 2: Find the vertex A Quadratic Function in Standard Form Thus the vertex is (1,–3). Since the x – value of the vertex is given by the line of symmetry, we need to plug in x = 1 to find the y – value of the vertex.

VERTEX FORM OF QUADRATIC EQUATION y = a(x - h) 2 + k  The vertex is (h,k).  The axis of symmetry is x = h.

Graphing Using the Vertex Form  Steps  Find the Vertex (h,k) & plot  Pick 2 values of x, <h  Substitute those values into the original function to find the y-coordinate & plot  Pick 2 values of x, >h  Substitute those values into the original function to find the y-coordinate & plot 13

GRAPHING A QUADRATIC FUNCTION IN VERTEX FORM (-3,4) (-7,-4) (-1,2) (-5,2) (1,-4) Axis of symmetry x y  Example y = -1/2(x + 3) where a = -1/2, h = -3, k = 4. Since a<0 the parabola opens down.  To graph a function, first plot the vertex (h,k) = (-3,4).  Draw the axis of symmetry x = -3  Plot two points on one side of it, such as (-1,2) and (1,-4).  Use the symmetry to complete the graph.

INTERCEPT FORM OF QUADRATIC EQUATION y = a(x - p)(x - q)  The x intercepts are p and q.  The axis of symmetry is halfway between (p,0) and (q,0).

GRAPHING A QUADRATIC FUNCTION IN INTERCEPT FORM (-2,0) (1,9) (4,0) Axis of symmetry x y  Example y = -(x + 2)(x - 4). where a = -1, p = -2, q = 4. Since a<0 the parabola opens down.  To graph a function, the x - intercepts occur at (-2,0) and (4,0).  Draw the axis of symmetry that lies halfway between these points at x = 1.  So, the x - coordinate of the vertex is x = 1 and the y - coordinate of the vertex is: y = -(1 + 2)(1 - 4)= 9.