5.1 – GRAPHING QUADRATIC FUNCTIONS (DAY 1) Algebra 2 Reward for sitting through 5.1 notes rehearsal

Objectives  Graph quadratic functions in standard form  Relate quadratic functions to real-life problems

 Quadratic function: A two-variable equation that has its greatest exponent on a variable raised to the 2 nd power  Form: Note: the variable “a” cannot equal 0. Why? What is a quadratic function?

Vocabulary  Parabola: a U-shaped graph of a quadratic function  Vertex: the highest or lowest point of a parabola  Axis of symmetry: the vertical line through the vertex of a parabola

Vocabulary axis of symmetry parabola vertex

standard form If an equation is in standard form… 1. Label a, b, and c 2. Find the vertex a) Use for the x-coordinate b) Plug x in to find the y-coordinate 3. Pick the next two integers greater than the x value of your vertex and find the corresponding y-values 4. Plot the ordered pairs 5. Use symmetry to plot two “mirror image” ordered pairs

Example 1 Vertex? Axis of symmetry? (2, -2) x = 2

Example 2 What is different about this graph compared to the others we’ve seen? Vertex? Axis of symmetry? (-3, -2) x = -3

Example 3 Vertex? Axis of symmetry? (-4, -3) x = -4

Exit Slip 1.Graph the following quadratic function. Identify the vertex and the axis of symmetry. 2.Rate your understanding of today’s lesson on a scale 1-5. (1 = I’m lost!  3 = I’m okay.  5 = This is easy!) Vertex? Axis of symmetry? (1, -4) x = 1

Homework/Reminders Due on Monday: pg. 253 #20-23 (make nice, accurate graphs) Quiz on Tuesday, November 20 th

5.1 – GRAPHING QUADRATIC FUNCTIONS (DAY 2) Algebra 2

Quadratic function

Objectives  Graph quadratic functions in vertex form  Graph quadratic functions in intercept form  Use quadratic functions to solve real-life problems

Vertex Form  An equation in vertex form is written as: Does this form look similar to anything we’ve done in the past?

Discovery Activity  Using your graphing calculator, find the vertex of the following functions. Try to identify a pattern. 1.y = 2(x – 1) y = -0.5(x – 2) 2 – 7 3.y = 3(x + 3) 2 – 1 4.y = 0.25(x – 1) 2 5.y = 2x (1, 6) (2, -7) (-3, -1) (1, 0) (0, 6)

Vertex Form  Equations in vertex form highlight the vertex of a quadratic equation. Vertex: (h, k)

vertex form If an equation is in vertex form… 1. Identify the vertex, (h, k), and plot the ordered pair 2. Pick the next two integers greater than the x value of your vertex and find the corresponding y-values 3. Plot the ordered pairs 4. Use symmetry to plot two “mirror image” ordered pairs

Example 1 Vertex? Axis of symmetry? (2, 6) x = 2

Example 2 Vertex? Axis of symmetry? (-3, 4) x = -3

Intercept Form  An equation in intercept form is written as: Examples: y = 2(x – 2)(x – 6) y = 4(x + 1)(x – 1) y = -(x + 8)(x + 7) How can these be quadratic functions if there is no exponent of 2 on a variable?

Intercept Form  Standard Form  Use FOIL to convert the quadratic equation in intercept form into a quadratic equation in standard form. y = (x – 2)(x – 6) y = 4(x + 2)(x – 1) y = -(x + 4)(x + 3) y = x 2 – 8x + 12 y = 4x 2 + 4x – 8 y = -x 2 – 7x – 12

Discovery Activity  Using your graphing calculator, find the intercepts of the following functions. Try to identify a pattern. 1.y = 2(x – 1)(x + 2) 2.y = -0.5(x – 2)(x + 4) 3.y =.25(x + 3)(x – 8) (1, 0) and (-2, 0) (2, 0) and (-4, 0) (-3, 0) and (8, 0)

Intercept Form  Equations in vertex form highlight the x- intercepts of a quadratic equation. x-intercepts: (p, 0) and (q, 0)

intercept form If an equation is in intercept form… 1. Find the x-intercepts (set each factor to zero and solve for x) and plot 2. Take the average of the x-intercepts to find the h of the vertex, and then find k, and plot. 3. Pick the next integer greater than the x value of your x-intercept and find the corresponding y- value 4. Plot the ordered pair 5. Use symmetry to plot the “mirror image” ordered pair

Example 3 x-intercepts? vertex? axis of symmetry? (-2, 0) and (4, 0) (1, 9) x = 1

Example 4 x-intercepts? vertex? axis of symmetry? (2, 0) and (-1, 0) (0.5, -9) x = 0.5

Homework/Reminders Due tomorrow: pgs #28, 30, 32, 36, 40, 42 (make nice, accurate graphs) Quiz on Tuesday, November 20 th Mechanical pencil