Vertex maximum or minimum Axis of Symmetry OPENS: UP MINIMUM

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1 5.1 GRAPHING QUADRATIC FUNCTIONS IN STANDARD FORM & MUTIPLYING BINOMIALS Standard Form of a Quadratic: y ax bx c or f x ax bx c ex. y x 5x 13 a= b= c=. Every function/graph in the Quadratic family originates from the parent function: y x. While we will not use a table to graph quadratics let s start off using the table to see what the quadratic parent looks like: y x x y What do they look like? What else are they called? PARTS OF A QUADRATIC GRAPH: Vertex maximum or minimum Axis of Symmetry VERTEX = ( ) OPENS: UP MINIMUM OPENS: DOWN MAXIMUM AXIS OF SYMMETRY:

2 Graph a quadratic function using a table y 3x. y x x y Vertex: Opens: up or down Vertex: Opens: up or down LET S GO INTO SOME DETAILS ABOUT THE PARTS OF A QUADRATIC! a Vertex When looking at the standard form for a quadratic its a value will tell you whether your graph will open up (think smiley face) or down (think frown). If a > 0 then the quadratic will open up. If a < 0 then the quadratic will open down. The Vertex is the lowest or highest point (x y) on a parabola/quadratic. If the vertex is the highest point on the graph we call it a maximum. If the vertex is the lowest point on the graph we call it a minimum. Because it is a point we need to know how to find the x and y coordinates. To find the x b coordinate we use the formula x (memorize) a Once you know the x coordinate of the vertex simply plug it back into the original equation and you will have the y coordinate of the vertex.

3 The axis of symmetry is an imaginary vertical line that divides the parabola into two equal parts. It only passes through one point on the entire graph the vertex. Axis of Symmetry c Since it is a vertical line its equation is of the form x = just like every vertical line. b The equation is x the same equation we just saw for the x coordinate of the vertex. So a the x coordinate of the vertex and the axis of symmetry will always be the same number!! When looking at the standard form for a quadratic its c value is the y-intercept. Remember the y- intercept is the point where the parabola or quadratic crosses the y-axis which takes the form (0 c). The purpose of finding c and writing it as a point is to help us find additional points on the graph! Graphing a quadratic WITHOUT a b value. (b = 0) Graphing a quadratic WITH a b value. (b = 6) 3. y x 6x a= b= c= 4. y x 4 a= b= c= a = Opens a = Opens c = c = Y-intercept Y-intercept Vertex Vertex Equation for Axis of Symmetry Maximum or Minimum Max/Min Value Equation for Axis of Symmetry Maximum or Minimum Max/Min Value

4 Directions: Fill in the following values to graph each quadratic. You will have to show all work for credit on tests and quizzes! 5. f x x 8x a= b= c= 6. x x 5 f a= b= c= a = Opens a = Opens c = c = Y-intercept Y-intercept Vertex Vertex Equation for Axis of Symmetry Maximum or Minimum Max/Min Value Equation for Axis of Symmetry Maximum or Minimum Max/Min Value

5 F O I L Multiply using FOIL. 4 x 6 1. x **. x 3 Multiply using the BOX method. 3 x 7 3. x **4. x Write a quadratic function in Standard Form. 5. y x 5 6. y x 1 3

6 5. GRAPHING QUADRATICS IN VERTEX FORM VERTEX FORM: y ax h k a : h : k : VERTEX: ( ) AXIS: Determine the following for each quadratic equation in VERTEX FORM. 1. y x 3 5. y x y x 3 4 a = h = k = a = h = k = a = h = k = VERTEX: VERTEX: VERTEX: AXIS: AXIS: AXIS: OPENS: OPENS: OPENS: MAX MIN MAX MIN MAX MIN VALUE: VALUE: VALUE: Y-INTERCEPT: Y-INTERCEPT: Y-INTERCEPT: REVIEW: Put the quadratic equation from #1 into standard form.

7 Given a quadratic equation in vertex form find the vertex axis of symmetry whether the graph opens up or down and find the maximum or minimum. Graph it! 4. y x 3 1 Vertex: Opens: up down Maximum Minimum Max/Min Value: y-intercept: 5. y x 3 Vertex: Opens: up down Maximum Minimum Max/Min Value: y-intercept: 6. y x 1 4 Vertex: Opens: up down Maximum Minimum Max/Min Value: y-intercept:

8 WRITE THE EQUATION OF EACH EQUATION IN VERTEX FORM: STEPS: 1. Find the vertex (h k). Substitute the vertex and a point on the parabola into the equation. 3. Solve for a. 4. Substitute h k and a into the vertex form for the equation. y = a(x h) + k EXAMPLES: Vertex: h = k = Fill in h and k into the equation: y = a(x h) + k Vertex: h = k = Fill in h and k into the equation: y = a(x h) + k Plug the ordered pair in from the graph for x and y that isn t the vertex. Ordered pair: Plug the ordered pair in from the graph for x and y that isn t the vertex. Ordered pair: