5.1 Quadratic Function 11/30/12

Graph is a parabola Vocabulary Quadratic Function : a function that is written in the standard form: y = ax 2 + bx + c where a ≠ 0 Vertex:The highest or lowest point of the parabola. Vertex the line that divides a parabola into mirror images and passes through the vertex. Axis of symmetry: Axis of symmetry

STEPS FOR GRAPHING y = ax 2 + bx + c Step 1: Find and plot the vertex. The x –coordinate of the vertex is Substitute this value for x in the equation and evaluate to find the y -coordinate of the vertex. Step 2: Draw the axis of symmetry. It is a vertical line through the vertex. Equation is x = # (x-coordinate of the vertex). Step 3: Make an x-y chart. Choose 2 (or more) values for x to the right or left of the line of symmetry. Plug them in the equation and solve for y. Step 4: Graph the points. Mirror the points on the other side of the line of symmetry. Draw a parabola through the points.

Graph y = x 2 ax 2 + bx + c where a = 1, b= 0 and c = 0 Simplest quadratic equation 1. Find the vertex: To find y, plug x in the equation and solve for y. Vertex: (0, 0) 2. Draw the line of symmetry: x=0 3. Make an x-y chart. Choose 2 (or more) values for x to the right or left of the line of symmetry. Plug them in the equation and solve for y. 4. Graph the points. Mirror the points on the other side of the line of symmetry. Draw a U-shaped curve through the points. xy x= 1, y = 1 2 y= 1 x = 2, y = 2 2 y = 4 Plot (1,1) & (2, 4)

Graph y = - x 2 Think y=-1x 2 where a = -1, b= 0 and c = 0 1. Find the vertex: To find y, plug x in the equation and solve for y. Vertex: (0, 0) 2. Draw the line of symmetry:x=0 3. Make an x-y chart. Choose 2 (or more) values for x to the right or left of the line of symmetry. Plug them in the equation and solve for y. 4. Graph the points. Mirror the points on the other side of the line of symmetry. Draw a U-shaped curve through the points. xy x= 1, y = y= -1 x = 2, y = y = - 4 Plot (1,-1) & (2, -4)

Graph of y = 1x 2 Graph of y = -1x 2 When a is positive, the parabola opens up. When a is negative, the parabola opens down.

Graph a Quadratic Function in Standard Form Example 2 Graph =x 2x 2 6x6xy5+ –.

Graph a Quadratic Function in Standard Form Example 2 Graph =x 2x 2 6x6xy5+ – SOLUTION The function is in standard form y ax 2 bx c where a 1, b 6, and c 5. Because a > 0, the parabola opens up. = ++ == – = STEP 1 Find and plot the vertex.. = 3x 2a2a b – = – 2 () 1 = 6 – =x 2x 2 6x6xy5+ – = 65+ – ()2)2 3 () 3 = 4 – The vertex is. () 3, 4 –

Graph a Quadratic Function in Standard Form Example 2 =x 2x 2 6x6xy5+ – =x 2x 2 6x6xy5+ – = 5+ – ()2)2 0 6 () 0 = 5 = 5+ – ()2)2 1 6 () 1 = 0 STEP 3Plot two points to the left of the axis of symmetry. Evaluate the function for two x -values that are less than 3, such as 0 and 1. STEP 2 Draw the line of symmetry. x=3

Graph a Quadratic Function in Standard Form Example 2 Plot the points and. Plot their mirror images by counting the distance to the axis of symmetry and then counting the same distance beyond the axis of symmetry. () 0, 5 () 1, 0 STEP 4Draw a parabola through the points.

Checkpoint Graph a Quadratic Function in Standard Form Graph the function. Label the vertex and the axis of symmetry. 4. = x 2x 2 6x6xy2 ––

Checkpoint Graph a Quadratic Function in Standard Form Graph the function. Label the vertex and the axis of symmetry. ANSWER 5. = x 2x 2 2x2xy1 –– +

Checkpoint Graph a Quadratic Function Using a Table Graph the function using a table of values. ANSWER 1. y = – 3 x 2x 2

Checkpoint Graph a Quadratic Function Using a Table Graph the function using a table of values. ANSWER 2. y = – x 2x 2 – 2

Homework 5.1 p.225 #20-25, 30-32, (5 graphs)

mathematical expressions with 2 terms. Ex. x + 2, 2x 2 -5, x Vocabulary Binomials: Multiplying Binomials: FOIL First, Outside, Inside, Last Ex. (x + 3)(x + 5) ( x + 3)( x + 5) (x + 3)(x + 5) = x 2 + 5x + 3x +15 = x 2 + 8x + 15 I O L First: x x = x 2 Outside: x 5 = 5x Inside: 3x = 3x Last: 35 = 15 F

Example 3 Multiply Binomials Find the product. () 3+2x2x () 7x – Write products of terms. SOLUTION () 7 – () 3+2x2x () 7x – = 2x2x () x+2x2x+3x3x+3 () 7 – = 2x 22x 2 14x+3x3x – 21 – Multiply. = 2x 22x 2 11x – 21 – Combine like terms. F LIO

Checkpoint Multiply Binomials Find the product. 7. () 4x – () 6x + ANSWER x 2x 2 +2x2x 24 – 8. () 1x – () 13x3x + ANSWER 3x 23x 2 2x2x 1 –– 9. – () 52x2x () 2x – ANSWER 2x 22x 2 9x9x 10 – +

Example 4 Write a Quadratic Function in Standard Form Write the function in standard form. Write original function. SOLUTION ()2)2 2x – y = 25+ ()2)2 2x – y = 25+ () 2x – = 25+ () 2x – Rewrite as. ()2)2 2x – () 2x – () 2x – () 2x2xx 2x 2 – = 24+2x2x – 5+ Multiply using FOIL. () 4x4xx 2x 2 – = Combine like terms. 8x8x2x 22x 2 – = 8+5+ Use the distributive property. 8x8x2x 22x 2 – = 13+ Combine like terms.

Checkpoint Write a Quadratic Function in Standard Form Write the function in standard form. 10. () 3x – () 1x + y = () 6x – y = 3 () 4x – 12. ()2)2 1x – y = 3 –– ANSWER 2x 22x 2 4x4x 6 –– y = y = 3x 23x 2 30x 72 – + ANSWER y = x 2x 2 2x2x 4 – + –