Warm Up Graph: 𝑦= 𝑥 2 +2𝑥−3 STEPS FOR GRAPHING y = ax2 + bx + c Step 1: Find and plot the vertex. The x –coordinate of the vertex is 𝒙= −𝒃 𝟐𝒂 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 of axis of symmetry 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.

5.2 Graphing Quadratic Functions in Vertex Form 11/13/13

Equations of a line… Slope intercept form: y = mx + b Ex: y = 3x + 2 Point – slope form: y - y1 = m(x – x1) Ex: y – 5 = 3(x – 1) Standard form: Ax + By = C Ex: 3x – 2y = 5

Equations of a Parabola… Standard Form: y = ax2 + bx + c Ex: y = 2x2 + 4x + 9 Vertex Form: y = a(x-h)2 + k

When a is positive, the parabola opens up. Vertex Form: a quadratic equation written in the form Y = a(x –h)2 + k (h, k) is the vertex Again… When a is positive, the parabola opens up. When a is negative, the parabola opens down. (h, k) Graphing Method: Step 1 Plot the vertex (h, k) from the equation Steps 2-4 : same as the standard form.

Example 1 Graph . ( )2 2 x – y = 1 + The function is in vertex form where , h 2, and k 1. Because a is negative, the parabola opens down. = a 2 – y k + ( )2 h x STEP 1 vertex. ( ) h, k 2, 1 = STEP 2 Draw the axis of symmetry through the vertex. STEP 3 x y 1 -7 -1 When x = 0 𝑦=−2 0−2 2 +1 =−2 −2 2 +1 =−2(4)+1 =−8+1=−7 When x = 1 𝑦=−2 1−2 2 +1 =−2 −1 2 +1 =−2(1)+1 =−2+1=−1 STEP 4 Plot (0, -7) and (1, -1). Mirror them on the other side of axis of symmetry. Draw a smooth curve through the points.

, h = 3, and k = -1. Because a is positive the parabola opens up. a = Example 2 ( )2 3 x – y = 1 Graph , h = 3, and k = -1. Because a is positive the parabola opens up. a = 1 STEP 1 vertex. ( ) h, k 3, -1 = STEP 2 Draw the axis of symmetry through the vertex. STEP 3 x y 4 5 3 When x = 4 𝑦= 4−3 2 −1 = (1) 2 −1 =0 When x = 5 𝑦= 5−3 2 −1 = (2) 2 −1 =4−1 =3 STEP 4 Plot (4, 0) and (5, 3). Mirror them on the other side of axis of symmetry. Draw a smooth curve through the points.

, h = -1, and k = 4. Because a is positive the parabola opens up. a = Example 3 Graph 4 y = ( )2 1 x 2 + , h = -1, and k = 4. Because a is positive the parabola opens up. a = 2 STEP 1 vertex. ( ) h, k -1, 4 = STEP 2 Draw the axis of symmetry through the vertex. STEP 3 x y 1 6 12 When x = 0 𝑦=2 0+1 2 +4 =2 1 2 +4 =2(1)+4 =2+4=𝟔 When x = 1 𝑦=2 1+1 2 +4 =2 2 2 +4 =2(4)+4 =8+4=𝟏𝟐 STEP 4 Plot (0, 6) and (1, 12). Mirror them on the other side of axis of symmetry. Draw a smooth curve through the points.

Maximum and Minimum Value of the Quadratic Eqn. Vertex is the highest point, therefore the y-coordinate of the vertex is the maximum value. Vertex is the lowest point, therefore the y-coordinate of the vertex is the minimum value.

Take y-coordinate as the minimum value of the function. Find the Minimum or Maximum Value Tell whether the function has a minimum or maximum value. Then find the minimum or maximum value. ( )2 8 x y = 12 – 2 1 + Find the vertex: (h, k) is (-8, -12) Since a is positive, the parabola opens up and the vertex is the lowest point. Take y-coordinate as the minimum value of the function. ANSWER minimum; 12 –

𝑦=− 𝑥 2 −2𝑥+1 Tell whether the function has a minimum or maximum value. Then find the minimum or maximum value. 𝑎=−1, b=−2, c=1 𝒙= −𝒃 𝟐𝒂 𝒙= −(−𝟐) 𝟐(−𝟏) = 𝟐 −𝟐 =−𝟏 𝑦= −1(−1) 2 −2 −1 +1=−1+2+1=2 Vertex (-1, 2) Since a is negative, the parabola opens down and the vertex is the highest point. Take y-coordinate as the maximum value of the function. ANSWER maximum; 2

Homework WS 5.2