Graphing Quadratic Equations in Vertex and Intercept Form Section 4.2 Graphing Quadratic Equations in Vertex and Intercept Form

The y intercept of the graph is c if the equation is in STANDARD FORM Quadratic Functions A quadratic function has the form: f (x) = ax2 + bx + c Where a, b and c are real numbers and a is not equal to 0. The y intercept of the graph is c if the equation is in STANDARD FORM . The basic shape of the graph is a PARABOLA or U shaped

Positive Quadratic Negative Quadratic y = ax2 y = -ax2 Parabolas always have a lowest point (minimum) or a highest point (maximum, if the parabola is upside-down). This point, where the parabola changes direction, is called the "vertex".

Vertex- Axis of symmetry- The lowest or highest point of a parabola. The vertical line through the vertex of the parabola. Axis of Symmetry

Vertex Form: y = a(x – h)2 + k, - the vertex is the point (h, k). - the axis of symmetry is x = h - if “a” is positive it opens up - if “a” is negative it opens down Plot the vertex and then find two other points by using “a” as your rise/run

Vertex Form (x – h)2 + k – vertex form Each function we just looked at can be written in the form (x – h)2 + k, where (h , k) is the vertex of the parabola, and x = h is its axis of symmetry. (x – h)2 + k – vertex form Equation Vertex Axis of Symmetry y = x2 or y = (x – 0)2 + 0 (0 , 0) x = 0 y = x2 + 2 or y = (x – 0)2 + 2 (0 , 2) y = (x – 3)2 or y = (x – 3)2 + 0 (3 , 0) x = 3

(-2, 5) EXAMPLE A Graph a quadratic function in vertex form 12 Graph y = – (x + 2)2 + 5. SOLUTION STEP 1: Identify the vertex (-2, 5) STEP 2: Plot the vertex & draw the line of symmetry STEP 3: Determine if it opens up or down

EXAMPLE A Graph a quadratic function in vertex form 12 Graph y = – (x + 2)2 + 5. STEP 4: Use “a” as your rise/run Down 1 and over 2 STEP 5: Draw a parabola

Example B: Graph y = (x + 2)2 + 1 Analyze y = (x + 2)2 + 1. Step 1 Plot the vertex (-2 , 1) Step 2 Draw the axis of symmetry, x = -2. Step 3 Find and plot two points on one side , such as (-1, 2) and (0 , 5). Step 4 Use symmetry to complete the graph, or find two points on the left side of the vertex.

Example C: Graph y=-.5(x+3)2+4 a is negative (a = -.5), so parabola opens down. Vertex is (h,k) or (-3,4) Axis of symmetry is the vertical line x = -3 Table of values x y -1 2 -2 3.5 -3 4 -4 3.5 -5 2 Vertex (-3,4) (-4,3.5) (-2,3.5) (-5,2) (-1,2) x=-3

Now you try one!  

Minimum “a” is positive Maximum “a” is negative GUIDED PRACTICE for Examples 1 and 2 Graph the function. Label the vertex and axis of symmetry. y = (x + 2)2 – 3 Minimum “a” is positive y = –(x + 1)2 + 5 Maximum “a” is negative

Intercept Form Equation y=a(x-p)(x-q) The x-intercepts are the points (p,0) and (q,0). The axis of symmetry is the vertical line x= The x-coordinate of the vertex is To find the y-coordinate of the vertex, plug the x-coord. into the equation and solve for y. If “a” is positive, parabola opens up If “a” is negative, parabola opens down.

Example D: Graph y=-(x+2)(x-4) Since a is negative, parabola opens down. The x-intercepts are (-2,0) and (4,0) To find the x-coord. of the vertex, use To find the y-coord., plug 1 in for x. Vertex (1,9) The axis of symmetry is the vertical line x=1 (from the x-coord. of the vertex) (1,9) (-2,0) (4,0) x=1

Now you try one! y=2(x-3)(x+1) Open up or down? X-intercepts? Vertex? Axis of symmetry?

x=1 (-1,0) (3,0) (1,-8)